THE 

UlsflVERSITY  OF  MISSOURI 

studies' 

MATHEMATICS   SERIES 

VOLUME  I        NUMBER  1 

ON  THE  DEFINITION  OF  THE  SUM 
OF  A  DIVERGENT  SERIES 

BY 

LOUIS  LAZARUS  SILVERMAN,  Ph.D. 

Instructor  in  Mathematics  in  Cornell  University 
Formerly  Instructor  in  Mathematics  in  the  University  of  Missouri 


UNIVERSITY  OF  MISSOURI 

COLUMBIA,  MISSOURI 
April,  1913 


ON  THE  DEFINITION  OF  THE  SUM 
OF  A  DIVERGENTiSERIES 


THE 

UNIVERSITY  OF  MISSOURI 
STUDIES 

MATHEMATICS    SERIES 

VOLUME  I        NUMBER   1 

ON  THE  DEFINITION  OF  THE  SUM 
OF  A  DIVERGENT  SERIES 

BY 

LOUIS  LAZARUS  SILVERMAN,  Ph.D. 

Instructor  in  Mathematics  in  Cornell  University 
Formerly  Instructor  in  Mathematics  in  the  University  of  Missouri 


UNIVERSITY  OF  MISSOURI 

COLUMBIA,  MISSOURI 

April,  1913 


Press  of 
The  New  era  printins  CoiiMN> 

LANCASTER,  PA. 


Miart.inatical 
Sciences 
Libra/y 


6<'A 


^   J; 


'^<^ 


CONTENTS 

Page 

§    I.  Introduction i 

§    2.  Historical  Resume 3 

§    3.  averageable  sequences i5 

§    4.  Product  Definitions 23 

§    5.  On  Certain  Possible  Definitions  of  Summability  33 

§    6.  Definitions  of  Evaluability 46 

§    7.  Applications 63 

§    8.  Tests  for  Cesaro-summability 76 

§    9.  Theorems  on  Limits 83 

§  10.  Conclusion 89 


3'167IM 


§  I .    INTRODUCTION  * 

The  series  Uo  +  Wi  +  «2  +  •  •  •  is  defined  to  be  convergent 
whenever  Ij  (wo  +  «i  +  •  •  •  +  Un)  exists;  and  the  value  of  this 

Hmit  is  called  the  swn  of  the  series.     If  this  limit  does  not  exist, 
the  series  is  said  to  be  divergent. 
Some  writers  call  a  series  divergent  only  when  Ij  («o+Wi+  •  •  • 

+«„)  =  00;  all  series  which  neither  converge  to  a  finite  limit 
nor  diverge  to  infinity  are  then  called  oscillatory,  f  The  present 
considerations  are  limited  to  series  which  are  oscillatory.  We 
shall  follow,  however,  the  terminology  of  most  writers |  by  calling 
divergent  all  series  which  do  not  converge;  stating  expressly, 
if  necessary,  when  a  series  diverges  to  infinity. 

A  necessary  condition  for  the  convergence  of  a  series  is  Jj  Wn  =  o. 

Thus  only  a  limited  number  of  series  can  be  dealt  with.  It  is 
accordingly  desirable  to  extend  the  definition  of  the  sum  of  a 
series,  so  as  to  include  a  larger  number  of  series  with,  which  we 
may  deal  rigorously.  Our  object  will  be  to  retain  the  class  of 
convergent  series,  and  to  add  to  that  set,  by  means  of  a  more 
general  definition,  as  large  a  class  as  possible  of  series  which 
are  not  convergent.  In  order  to  be  able  to  deal  with  these 
new  series,  however,  we  shall  wish  to  preserve  several  funda- 
mental properties  of  convergent  series.  We  shall,  in  fact, 
demand  the  following  fundamental  requirements  of  any  general- 
ized definition  of  the  sum  of  a  series: 

*  This  paper  was  accepted  as  a  dissertation  by  the  Graduate  Faculty  of 
the  University  of  Missouri  in  May,  1910,  in  partial  fulfillment  of  the  require- 
ments for  the  degree  of  Doctor  of  Philosophy. 

t  Bromwich:  An  Introduction  to  the  Theory  of  Infinite  Series,  p.  2. 

t  See  e.  g.,  Goursat-Hedrick:  Mathematical  Analysis,  p.  327. 


INTRODUCTION 

(i)  The   generalized   sum   must  exist,   whenever  the  series 

converges, 
(ii)  The  generalized  sum  must  be  equal  to  the  ordinary  sum, 

whenever  the  series  converges, 
(iii)  Each  of  the  series 

fwo  +  Wi  4-  W2  +  •  •  • 
«i  +  «2  +  •  •  • 

has  a  generalized  sum,  whenever  the  other  has,  and 
t  =  s  —  uq,  [{  s  and  t  are  their  respective  sums, 
(iv)  If  each  of  the  series 

uo  +  Ui  +  U2  -\-  •  •  • 
vo  -}-  Vi  -\-  V2  -\-  •  •  • 

has  a  generalized  sum,  A  and  B  respectively,  then  the 

series   («o  +  z'o)  +  (wi  +  Vi)  +  (w2  +  ^'2)  +  •  •  •    has  a 

generalized  sum  which  is  ^4  -\-  B. 
(v)   If  the  series  Uo  +  Ui  -\-  U2  -\-  •  •  •  has  5  for  its  generalized 

sum,    then   ktio  +  kui  +  •  •  •    has   a   generalized   sum 

which  is  ks. 
I  wish  to  express  my  gratitude  to  Professor  E.  R.  Hedrick 
for  his  interest  in  my  work,  and  to  acknowledge  my  indebted- 
ness to  him  for  many  helpful  and  important  suggestions.  I  am 
also  indebted  to  Drs.  W.  A.  Hurwitz  and  H.  M.  Shefifer  for 
many  suggestions  and  criticisms. 


§  2.    HISTORICAL   RESUME  * 

The  earliest  interest  in  divergent  series  centers  about   the 

series 

I  -  I  +  I  -  I  +  •••. 

If  we  assume  that  this  series  has  a  generalized  sum  s,  then  the 
series,  obtained  by  dropping  the  first  term,  —  i  +  i  —  i  +  i''- 
must,  by  the  third  fundamental  requirement  of  page  2,  also 
have  a  generalized  sum  which  is  obviously  —  s.  We  have  then, 
s  —  i  =  — sors  =  ^.  Thus,  if  the  series  is  to  have  any  value 
at  all,  that  value  must  be  |.  And  this  is  precisely  the  value 
which  Leibniz t  was  led  to  attach  to  the  series,  by  different  con- 
siderations. The  sum  of  n  terms  of  the  series  is  o  or  i  according 
as  n  is  even  or  odd;  and  since  this  sum  is  just  as  often  equal  to  i 
as  it  is  to  o,  its  probable  value  is  the  arithmetic  mean,  ^.  This 
same  value  was  later  attached  to  the  series  by  Euler,  J  in  a  more 
satisfactory,  though  not  entirely  rigorous  manner.  "  Let  us 
say  that  the  sum  of  any  infinite  series  is  the  finite  expression, 
by  the  expansion  of  which  the  series  is  generated.  In  this 
sense,  the  sum  of  the  infinite  series  i  —  x  -\-  x^  —  x^  • '  •  will 
be  i/(i  +x),  because  the  series  arises  from  the  expansion  of 
the  fraction,  whatever  number  is  put  in  place  of  x."^  In  par- 
ticular, 

i  =  I  -  I  4-  I  -I  +  ••-. 


*  The  best  historical  sketches  are  to  be  found  in  Borel:  Legons  sur  les 
Series  Divergentes:  Introduction,  and  in  an  article  by  Pringsheim  given  im- 
mediately below. 

t  See  Pringsheim:  Encyclopddie  der  Math.  Wiss.,  I,  I,  p.  107,  note. 

XInstit.  Calc.  Diff.  (1755),  Paris,  II  (p.  289). 

§This  quotation  is  taken  from  Bromwich,  loc.  cit.,  p.  266. 

3 


4  UNIVERSITY   OF   MISSOURI   STUDIES 

It  is  true,  as  has  already  been  intimated,  that  none  of  the 
methods  given  above,  to  prove  that  the  series  should  have  the 
value  I,  is  satisfactory  from  a  theoretical  point  of  view.  But 
objections  have  been  raised*  to  the  result  for  practical  reasons 
also.  Thus,  the  series  i—  i  +  i  —  i  +  ---  may  be  obtained 
from  the  expansion 

I   -\-  X  I    —  v2 


I   +  X  +  X^         I 

and  setting  x  =  i, 


=  I  —  x^  -\-  x^  —  x^  -\-  X*'  —  x^  -{- 


1  =  1-1  +  1-1  +  •... 

To  meet  this  difficulty,  Lagrangef  observed  that  we  should  write 

; ; ^  =l-\-0'X—  X--\-X^-hO'X'^—  X^  -{-'•-, 

I  -\-  X  -]-  x^ 

so  that  for  x  =  i,  we  have 

f  =  i+o-i  +  i+o-i  +  ---. 

If  we  now  follow  the  method  of  Leibniz,  we  see  that  the  sequence 
corresponding  to  this  series  has,  out  of  every  three  succeeding 
terms,  once  the  value  o  and  twice  the  value  i ;  its  sum  is  accord- 
ingly 3-  Thus,  Lagrange  has  removed  the  practical  objection. 
Moreover  the  above  method  has  been  put  on  a  rigorous  theoretical 
foundation,  by  means  of  the  following  proposition,  J  which  is  a 
generalization  of  Abel's  theorem: 

Theorem  a:§  //  5„  =  wo  +  th  +  «2  +  •  •  •  +  «n  and 

^0  +  -^1  +  •  •  •   +  ^n 


W  +  I 


=  s, 


*  By  Callet.     See  reference  immediately  below. 

t  Rapport  sur  le  Memoire  de  Callet,  in:  Memoires  de  la  classes  des  Sciences 
mathematiques  et  physiques  de  VInstitut,  t.  III. 

X  Frobenius:  Journal  de  Crelle,  t.  89,  p.  262. 

§  Theorems  embodying  new  results  we  shall  indicate  by  numerals;  all  other 
theorems  will  be  lettered  A,  B,  C,  •  •  •. 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  5 

then 

n 

L  53  UnOC"^    =   S. 
z=l     0 

Thus,  in  the  case  of  the  series  i  —  i+i  —  !  +  •••, 

T   ^0  +  ^1  +  •  •  •  +  ^n  _  1 
•^  n  ""2' 

and    accordingly    Ij(i— Jt^  +  x^  •••)=!;    so    that    we    may 

define  the  value  of  the  series  i  —  i  +  i*--    tobeIj(i— :>;: 

-{-  x^  —  x^  -\-  •  •  •)>  or  what  amounts  to  the  same  thing, 

-y    5o  +  ^1  +  •  •  •  +  ^n 

whenever  the  limit  exists. 

The  first  mathematician  actually  to  carry  through  the  de- 
finition was  Ces^ro,*  who  approached  the  subject  from  another 
standpoint.     Cauchy  has  defined  as  the  product f  of  two  series 

f  Wo  +  Wi  +   •  •  • 

the  series 

UqVq  +  (wot'i  +  tiiVQ)  +  (2/0^2  +  UiVi  +  U2Vq)  +  •  •  • ; 

this  definition  being  justified  by  the  theorem,  due  also  to  Cauchy, 
that  the  product  series  thus  defined  of  two  absolutely  convergent 
series,  is  itself  absolutely  convergent.  Mertenst  has  generalized 
this  theorem  by  proving  that  the  Cauchy  product  of  an  abso- 
lutely convergent  series  by  a  simply  convergent  series  is  con- 
vergent. The  product  of  two  simply  convergent  series  may, 
however,  be  divergent.  Ces^ro  has  studied  the  divergent  series 
which  result  from  the  product  of  two  simply  convergent  series, 
and  has  obtained  the  following  remarkable  theorem: 

*  Bulletin  des  Sciences  maihematiques,  t.  XIV,  1890. 
t  We  shall  later  refer  to  this  as  the  Cauchy-product. 
X  Journal  de  Crelle,  t.  79,  p.  182. 


6  UNIVERSITY   OF   MISSOURI   STUDIES 

Theorem  b  :  Let  the  two  series 

I  «o  +  Wi  +  «2  +  •  •  • 

converge  to  u  and  v  respectively,  and  let 

W„   =   (UoVn  +  UiVn-l  +    •  •  •    +  n„Vo) 
Sn   =   IVo  -\-  Wi  +    ■  ■  •    +  Wn 


then 


LSo  -\-  Si  -\-  • '  •  -\-  Sn 
^— =  II  ■  V. 


n  -\-  I 

The  two  theorems  which  we  have  stated  justify  us  in  stating 
the  following  definition: 

Definition:*  //  5„  =  wo  +  ^i  +  «2  H —  •  +  Un,  the  series  Uq  +  iii 
-\-  •  •  •  -\-  Un  -^  •  •  •  is  summahle  and  has  the  valne  s  whenever 

LSo  -\-  Si  -\-  •  •  •  ~\-  Sn 
. =  s. 

n  +  I 

Let  us  now  proceed  to  show  that  this  definition  satisfies  the 
fundamental  requirements  of  page  2.  To  this  end,  we  shall 
prove  the  following  theorems. 

Theorem  c:'\  If  a  series  converges,  it  is  summahle,  and  the  two 
definitions  give  the  same  sum. 

Let  Sn  =  Uq  -\-  Ui -{•••'  -\-  Un,  and  L  5„  =  5;  we  shall  prove 

n=co 

that 


L-^O  +  ^1   +    •  •  •    -\-  Sn 

n=«  W   +    I 


We  have: 

5o  +  -Jl  +    •  •  •    -\-  Sn 


W  +   I 
(50-5)  + (51-5) -I |-(5g  — 5)  +  (5g+i-5)H |-(^ri-5) 


n  +  I 

|5o-5|  +  |5i-5|-| i-kg-l--y|  |5g-5|H hl^n--^ 

""  «  +   I  W  +   I 


*  Ces^ro  calls  series  of  this  type  sitnply  indeterminate. 
t  By  this  theorem  requirements  (i)  and  (ii)  are  satisfied. 


DEFINITION   OF  SUM   OF  A  DIVERGENT   SERIES  7 

Since  Tj  Sn  =  s,  we  can  take  q  so  great  that  |  5,  —  5  |  <  e  /  2, 

i>  q.     Having  chosen  this  q,  let  L  be  the  largest  of  the  numbers, 
\si  —  s\,  i  =  o,  I,  2,  ' '  •  q  —  I.    Then  we  obtain: 

^0  4-  5i  +  •  •  •  +  5„  .     qL      An  -  q-\-  i)e         qL         e 

, —  5    S ; + } ; ^ —  < 1 +  - 

W  +  I  w  +  I  2{n  +  l)  W  +  I        2 

We  can  now  choose  n  so  large,  n  >  r,  that 

gL         e 

< 


n  +  I      2 
and  hence, 

■Jo  +    ^1   +    •  •  •    +   5n 

—  s\  <  e,     n  >  r. 


w  +  I 


nkL  n+i  J- 


I 


=  s. 

Theorem  d:*  Each  of  the  series 

Wo  +  wi  +  «2  +  •  •  • 

Ui  +  W2  +   •  •  • 

is  summable  when  the  other  is;  and  s  and  t,  their  respective  sums, 
are  connected  by  the  relation  s  —  uq  =  t. 

We  shall  prove  only  one  part  of  this  theorem,  the  method  for 
the  second  part  being  exactly  the  same.  We  begin  by  proving 
the  following  fact. 

Lemma:  If  the  sequence  Sq,  Si,  •  •  •  5„,  •  •  •  is  summable  and  has 
5  for  its  sum,  then  the  sequence  Si,  Si,  •  •  •  Sn,  •  •  •  is  also  summable, 
its  sum  being  likewise  5. 

For, 

L-^i  +  ■^2  +  •  •  •   +  Sn+l        -J-        So       ,     T    ^^  '^  '  ' '  '^  •^"+^ 
__ =  ^  _l_  ^  — -— 

TC=oo  n  ~f~  I  71=00  n  ~f~  I       n=:«)  n  ~\~  i 

_    J    ^0  +  •?!  +    •  •  •    +  Sn+l        J     So  +  Si  +    •  •  •    +   Sn+l      W  +   2 


«  +  I  n=«  n  -\-  2  n  -\-  I 


LSa+   Si   -\ -\-  Sn 

«=«  W  +   I 


By  this  theorem  requirement  (iii)  is  satisfied. 


8  .  UNIVERSITY  OF   MISSOURI    STUDIES 

To  return  now  to  Theorem  d;  we  wish  to  prove  that  if 
«o  +  i'l  +  W2  +  •  •  •  is  summable  to  s,  then  Ui  +  «2  +  •  •  • 
is  summable  to  5  —  iiq.  The  sequence  corresponding  to  the 
series  Wo  +  Wi  +  W2  +  •  •  •  is  «o,  Uo  -\-  Ui,  •  •  • .  By  the  lemma 
proved  above,  it  follows  that  the  sequence  Mo+«i,  Uo-\-Ui-\-U2,  •  •  • 
or  Si,  52,  •  •  •  is  summable  to  s.  The  sequence  corresponding  to 
Ui  +  «2  +  •  •  •  is  Ui,  III  +  W2,  •  •  •  which  may  be  written 
5i  —  Uq,  52  —  uo,   •  •  ••     Now 

J      r  (51   —   Mo)    +   (52   —  Uq)   +   •  •  •    +   (5n   -   lip)  1 
n=ao  \_  n  \ 

-r      /5i  +  52  +    •  •  •    +  5n  \ 

=   _Li    I    —    «0    I    =   5   —   Mo. 

n=«  \  n  J 

Theorem  e:*  If 

Uo  +  Ui  +  •  •  • 
vo  +  Vi  -\-  •  • - 

are  summable  to  u  and  v  respectively,  then  the  series  (mq  +  ^o) 
+  (mi  +  z'l)  +  •  •  •  is  summable  to  u  -\-  v. 

Writing  Sn  =  Uo  +  Ui  -{-••'-{-  Un,  tn  =  Vq  +  Vi  +•  -  •+  Vn, 
we  have  5„  +  ^„  =  (mq  +  Vq)  +  (mi  +  z;i)  +  •  •  •  +  (m„  +  »„)• 
We  obtain: 

y       (50   +  /q)    +    (5i    +   /l)    +    •   •  •    +    (5n    +   /„) 

n=oo  W   +    I 

T     50  +  5i  +   •  •  •    +  5„          -r     /o  +  /l   4-    •  •  •    +  /n 
=   Li  7— +  Ij    T— =   M  +  ZJ. 

«=«  W   +   I  „=»  W  +    I 

Cesaro's  definition  of  summability  has  accordingly  been  justified 
from  the  theoretical  standpoint  of  our  requirements  for  any 
generalized  definition.  We  may  naturally  ask  the  practical 
question:  how  large  is  the  class  of  series  with  which  this  defi- 
nition enables  us  to  deal?  A  partial  answer  to  this  question  is 
contained  in  the  following  proposition: 

*  By  this  theorem  requirement  (iv)  is  satisfied.     See  also  note  p.  19. 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES  9 

Theorem  f:  A  necessary  condition  for  the  summability  of  the 
series  Uo  -\-  Ui  -{■•'•  -\-  Un  •••  is 

JL  —  =  o. 

»=«  n 

Since  the  series  is  summable, 

5o  +  5i   +    •  •  •    +  Sn-l  T     ^0  +  -Jl   +    •  •  •    +  >^n 

^ n ^ — ;r+"i —  ^"^ 

_    -y     -^0  +  -yi   +    •  •  •    +  Sn-l  _   J     So  -{■  Si   -\-    •  •  •    -{-  Sn 


Hence : 


LUfi           -w-      On           S  n—\  -r      Sn  -w-      Sn  —  1 

—  =  Li =  L  -  -  Li  -—  =  o. 


We  are  accordingly  limited  to  series  for  which 

(I)  L  -  =  o. 

n=(io   W 

But  such  a  simple  series  as  i— 2  +  3  —  4  +  5---  fails  to 
satisfy  this  condition.  Furthermore,  this  series  can  be  easily 
evaluated  by  following  out  the  principle  of  Euler;  for  if  we  put 
X  =  I  in  the  expansion: 

T 

=     I     —    2X    -\-    TyX^    •    •    •  , 


(I  +  xY 
we  obtain 

?  =  i-2  +  3-4+---. 

We  are  thus  led  to  extend,  with  Cesaro,  the  above  definition  of 
summability  of  order  i,  to  summability  of  order  2.  We  say 
that  a  series  is  summable  of  order  2,  if 

J      {n   +    l)-yo  +  ^^1   +    •  •  •    +   2Sn-\   +  Sn   _ 

h  {n  +  i){n  +  2)  ~'- 


10  UNIVERSITY  OF  MISSOURI   STUDIES 

A  necessary  condition*  for  the  existence  of  this  limit  is  that 

Ltifi 


so  that  we  cannot  evaluate  the  series, 

r(r  +  i)  _  r(r  +  i)(r  +  2) 
2!  3 

although  we  obtain  by  Euler's  method, 


.     'V.'       \^   ^)  '  \'  '■)\>       \      ■^ >      ,  ^ 

I-.  +  — :^ -, +•••,    r>2, 


r{r  +  i)    „       rij  +  \){r  +  2) 
=  I  —  rjc  + i —  :x;2  — — i ^   + 


(i  ■\-xy  2! 

and  accordingly 

I  r{r  -\-  i)       r(r  +  \){r  +  2) 

We  are  thus  led  to  state  the  following  more  general  definition : 

Definition  rf  The  series  Mo  +  Wi  +  M2  +  •  •  •   is  summable  of 

order  r,  if  r  is  the  smallest  integer  for  which  there  exists  the  limit: 

r{r-\-\)' • '{r-\-n—\)         r{r-\-\)- "{r-\-n  —  2) 
''  ^!  +'^  in-i)\  +"• 

r{r+i) 

^^^    n-k  (r+i)(r+2)--.(r+n)  ' 

w! 

This  definition  includes  convergence  for  r  =  o;  it  also  includes 
the  other  definitions  given  above  for  r  =  i,  2  respectively. 
We  shall  not  prove  that  this  definition  satisfies  the  requirements 
of  page  2;  this  is  easily  verified.  J 

Let  us  now  return  to  Ceskro's  first  definition,  and  observe  that 
we  may  generalize  it  in  a  more  natural  way. 


*  Bromwich,  loc.  cit.,  p.  318. 

t  Ces^ro,  loc.  cit. 

I  This  is  done  in  a  more  general  case,  infra,  pp.  55-57. 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  II 

Definition:*  Let 

5o  +  5i  +    •  •  •    +  5„ 


(3) 


/    (1)    = 


W   +    I 


I    (r+1)    _  r    =    T       O 

tn  ^^    J  ,        r  I,    2, 


then  the  smallest  integer  r  for  which  L  tn^""^  exists,  shall   make 

the  series  summable  of  order  r. 

To  distinguish  this  definition  from  that  on  page  lo,  we  shall 
call  the  definitions  Ces^ro-summability  of  order  r  and  Holder- 
summability  of  order  r,  denoting  them  briefly  by  {Cr)  and  {Hr) 
respectively.  It  is  knownf  that  these  two  definitions  are  equiv- 
alent for  the  same  r. 

We  may  now  ask  how  big  a  class  of  series  this  generalized 
definition  enables  us  to  deal  with.     If  a  series  is  {Cr),  then  J 

n=a>  n 

Accordingly  the  series   i  —  t  -\-  f  —  t^  -\-  •  •  •  {t  >  i)   does  not 
have  a  sum  (Cr)  for  any  value  of  r]  since 

L-  +  0,   /  >  I. 

n=oo  "■ 

We  are  thus  led  to  generalize  still  further  the  definition  for  the 
sum  of  a  series. 

From  the  definition  given  on  page  lo,  it  is  clear  that  we  may 
write  Cesaro's  forms  as  follows: 

_     T      ["gp^O  +   aiSl   +    •  •  •    +  CLnSnl 
n=z«  L         Oo  +  fll   +    •  •  •    +  On         J  ' 


*  Holder:  Mathematische  Annalen,  Bd.  20,  p.  535. 
t  Schnee:  Math.  Annalen,  Vol.  LXVII  (1909),  p.  no. 

Ford:  Am.  Journal  of  Math.,  Vol.  XXXII  (1909),  p.  315. 
X  Borel,  Series  divergentes,  p.  92. 


12 


UNIVERSITY   OF  MISSOURI    STUDIES 


where  the  Ui  are  functions  of  both  n  and  r,  r  being  fixed.*     Let 
us  choose  as  our  definitionf 

io{r)sQ  + 


=  L 


r=«>   n= 


J    V  ao(r)so  +  aiir)si  +  •  •  •  +  an{r)sn  "| 
n=«  L      ao(r)  +  fli(r)  +  •  •  •  +  flnW      J 


ao(r)  + 
In  particular  we  shall  take  ap{r)  =  r^/pl,  and  obtain 


=  L  L 


(4) 


r=cc  n=^ao 


I  2 !  « ! 

I       2 !  n ! 


=  Li   Jj  e-'  \  So  -\-  Si-+  •  "  +  Sn  — 

It  can  be  proved  readily |  that  this  limit  exists,  whenever  the 
series  converges.     We  shall  now  transform!  this  limit. 
Let  II 


s(r)  =  So  +  Si~  +  S2—,+  •  •  •  +  ^n  — j  + 

n ! 


S'{r)    =    5i   +  52  -  +   53  -j  +    •  •  •    +  Sn+l  — ,  + 
12!  Ill 


then 


ui(r)  =  s'{r)  -  sir)  =  Ui  +  «2-  +  z'2— j  +  •  •  •  +  u„  —  + 

I  2  '  ^^ 

But 


w! 


rfr 


[e-^5(r)]  =  e-'[s'{r)  -  s(r)], 


so  that 


and 


e-'Sir)  =    I    e-q5'(r)  -  s{r)]dr  +  «o 


r)dr. 


*  Borel,  Series  divergenies,  p.  94. 
t  r  is  now  a  positive  real  number. 

X  Bromwich,  loc.  cit.,  p.  298.     This  is  a  special  case  of  Th.  12,  p.  52  (infra). 
§  Borel,  loc.  cit.,  p.  97. 

II  It  is  assumed  that  s{r)  is  convergent  for  all  values  of  r;  otherwise  the 
limit  (4)  would  have  no  meaning. 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES  1 3 

If  now  we  integrate  by  parts  we  obtain: 

5  -  «o  =     e-"-  1    ui{r)dr       +1     «~'      I    Ui{r)dr    dr, 
or,  if  we  let: 
u{r)  =  Mo  +  u\r  +  "2 -,+  ■••+  «;j—  H-  ••  •  =  Mo  +  I    ui(r)dr, 

2  .  71  ,  ^Q 

e-''[u{r)  —  uo]dr 
=  [e-''ii{r)]o  —  Mo[e~ir  +   I     e-''u{r)dr  -  Uq  I     g-'-f/r 

t/Q  Jo 

=  [e-''u(r)]o  +   I     e~''u{r)dr, 


1.  e., 

5  —  Mo 

or 


Ij  [e-''M(?')]  —  Mo  +  I     e-Hi{r)dr, 

r—'n  i/O 

e-'"M(r)(/r. 

e~''ii{r)dr  is  convergent,  then  it  follows 

from  the  last  equation  that  L  [e~''u(r)]  must  exist.     But  this 

limit  must  necessarily  be  zero,  for  otherwise,  the  integral  would 
not  converge.     Hence  we  obtain 


s  =    j     e~''u(r)dr, 


(5) 

where 

whenever  the  integral  converges.     It  can  be  proved f  here,  too, 


u{r)  ^  Mo  +  Ml  -  +  M2  ;7j  +  •  •  •  +  «n  — ,  + 


*  We  have  gone  into  greater  detail  here  than  does  Borel,  loc.  cit.,  p. 
But  this  is  essentially  his  argument, 
t  Bromwich,  loc.  cit.,  p.  269. 


14  UNIVERSITY   OF   MISSOURI   STUDIES 

that  when  the  series  «o  +  «i  +  •  •  •  +  «n  +  •  •  •   converges,  so 
docs  the  above  integral,  and  their  values  are  the  same. 
Furthermore  Borel  proves  the  following  theorem: 
Theorem  g  :*  //  the  Borel-integral  definition]  applies  to  the  series : 

«1   +  «2  +    •  •  •    +  Wn  +    •  •  •     =   S, 

then  it  also  applies  to  the  series  Mo  +  «i  +  «2  +  •  •  • ,  giving  for 
its  sum  s  +  2/0- 

The  converse,  however,  is  not  necessarily  true.  Thus  if  the 
series  Uo  -\-  Ui  -\-  th  -\-  •  •  •  is  summable  by  (5),  it  does  not  follow^ 
that  the  series  Ui  +  z<2  +  •  •  •  is  summable  by  (5).  Since  this 
fact  is  opposed  to  the  requirement  (iii),  page  2,  we  are  led  to 
modify  the  above  integral  definition,  and  to  state,  with  Borel, 
the  following  generalization: 

Definition:  The  series  Uq  -\-  Ui  -\-  ih  -\-  •  •  •   shall  be  called  ab- 

solutely    summable,     whenever    the    integrals      I     e~''  |  u{r)  \  dr, 

Jo 


[ 


e'"  I  u''^\r)  I  dr   converge,   where   X   denotes   the   order   of  any 


derivative. 

That  this  definition  satisfies  requirement  (Hi)  is  proved  by  the 
following  theorem :  § 

Theorem  h  :  If  either  of  the  series 

Uq  +  «1   +  «2  +    •  •  • 
Wi  +  ^2  +    •  •  • 

is  absolutely  summable,  so  is  the  other;  and  if  s,  t  be  their  respec- 
tive values,  we  have  s  —  Uq  =  t. 

We  shall  not  enter  into  the  further  generalizations  which 
have  been  given  by  Borel  himself  and  by  Le  Roy.|| 

*  Borel,  loc.  cit.,  p.  loi. 

t  We  shall  call  the  two  definitions  given  by  Borel,  the  Borel-mean  and  the 
Borel-integral  definition  respectively. 

X  For  an  example,  see  Hardy,  Quarterly  Journal,  Vol.  35  (1903),  p.  30. 

§  Borel,  loc.  cit. 

II  Le  Roy:  Annates  de  la  Faculte  de  Sciences  de  Toulouse  (2°  series),  t.  2 
(1902),  p.  317.     See  p.  60,  footnote. 


§  3-    AVERAGEABLE    SEQUENCES 

On  page  4  we  have  considered  the  series 
I  -  I  +  I  -  I  +  •  •  • 


I+0-I  +  I+0-I  +   ---, 

and,  replacing  them  by  their  respective  sequences,  we  obtained 


i  =  I,  o,  I,  o,   ••• 
f  =  I,   I,  o,   I,   I,  o, 


The  probability-method  of  Leibniz*  consists  in  taking  for  the 
sum  of  the  sequence,  the  average  of  its  limit-values.  This  method 
has  been  justified  by  the  theorems  of  Frobeniusf  and  Cesaro,J 
and  the  further  generalizations.  We  propose  now  to  give  a 
justification  of  the  method  from  another  point  of  view. 

To  define  the  sum  of  a  sequence  as  the  average  of  its  limit-values 
is  obviously  not  adequate;  for  although  we  can  tell  that  the  limit 
I  is  to  be  counted  twice  in  the  sequence  considered  above, 

I,  I,  o,  I,  I,  o,   • ••, 

it  is  not  easy  or  even  possible  to  state  the  multiplicity  of  the 
limit-values  in  general,  as  is  evident  from  the  following  example: 

Si  =  o,  i  ^  n^  ]      _ 

So,   Si,    S2,     '  •  '  Sji,    '  '  '  ,      •  o  f  ^  —  O,  I,  2,  •  •  •. 

Si  =  I,  I  =  n-  ] 

To  meet  this  difficulty,  we  shall  proceed  as  follows. 
Let  us  assume,  to  be  concrete, §  that  the  sequence 

•^Ot    •^l)    -^2)     *  ■  ■    -^n)     *  '  ■ 


*  See  page  3. 
t  See  page  4. 
X  See  page  5. 

§  We  shall  go  into  every  detail  in  only  this  simple  case;  the  later  general- 
izations we  shall  outline  only  briefly. 

15 


l6  UNIVERSITY   OF  MISSOURI    STUDIES 

has  two  limit-values  h  and  h.     Then  we  have 

\sm  -  /il    <  e,      \s„  -  l2\<  e, 

for  an  infinite  number  of  values  of  m  and  of  n,  provided  ni,  n>N. 
Having  chosen  e  and  N,  let  us  now  choose  i  >  N;  then  there  will 
be  m  of  these  i  numbers  5,-  which  fall  in  the  interval  about  h, 
and  n  which  fall  in  the  interval  about  h-  Since  w  and  n  are  func- 
tions of  i,  we  may  write  m  =  fi{i),  n  =  f^ii).  If  we  choose  e  suffi- 
ciently small,  and  i  >  N,  we  shall  have 

fid)  +/2«  +k  =  i, 

where  ^  is  a  constant  independent  of  i. 

Definition:  The  sequence  5c,  Si,  S2,  ■  •  •  Sn,  •  •  •,  having  h  and  h 
as  limit-vahies,  shall  be  called  averageahle  and  have  s  for  its  sum 
provided 

•    itil  h{i)+h{i)  J    '• 

That  this  limit,  when  it  exists,  does  not  depend  upon  the 
particular  e  we  have  chosen  follows  at  once.  For  if  we  take 
e  <e,  calling  the  corresponding  functions  /i(i)  and  fiii),  it  is 
clear  that 

hii)  =  7i«  +  h 
h{i)  =  hii)  +  k2 

where  ^1,  k^  are  independent  of  i.     We  accordingly  have: 

[/i(i)  -  ^i]/i  +  [hii)  -  k^lh 


i^A   Mi)-\-f2(i)   i 


^  h  I  Uiii)  -  H  +  u,{i)  -  H 
=  L 


fi{i)  —  ki  /'(*)  ~  ^2 


ciV  hii)  +  hii)  J' 


DEFINITION    OF    SUM    OF    A    DIVERGENT   SERIES  1 7 

since 

T      ^1    _  X      ^2    _ 

Let  us  now  find  the  sum  of  the  sequence  suggested  on  page  15, 
I ;  o,  o;  I,  o,  o,  o,  o;  1,0,0,0,0,0,0;   •  •  • , 


1.  e., 


5i  =  I ,     i  =  n-  1 
m2  J 


o,     i  =# 


Let  us  choose  i  =  m,  and  let  n^  be  the  largest  square  integer 
less  than  or  equal  to  m.     Then  we  have: 

-w-    n  ■  I  -\-  (m  —  n)  ■  o       -^    n 
s  =  Li  =  ij  -  =  o, 

since  W"  ^  m. 

Let  us  now  see  whether  this  definition  satisfies  the  require- 
ments of  page  2.  The  first  two  requirements  are  obviously 
satisfied.  As  to  the  third,  we  observe  that  corresponding  to 
the  series  ^0  +  ^1  +  ^2+  •  •  •  +  w„  +  •  •  • ;  Wi  +  ^^2  +  •  •  •  +  w„ 
+  •  •  • ,  we  have  the  sequences  So,  Si,  5?,  •  •  •  5„,  •  •  • ;  Si  —  jcq,  s^  —  Ho, 
• '  •  Sn  —  iio,  •  •  • ;  and  if  the  limit-values  of  the  first  sequence, 
which  will  be  assumed  to  be  averageable  to  s,  be  h  and  h,  then 
those  of  the  second  sequence  are  h  —  Uq,  h  —  Uq.  We  accord- 
ingly have: 


-iio  =  s—Uo. 


We  shall  now  show  that  the  fourth  requirement  is  satisfied. 

Theorem  i  :     The  sum  of  two  averageable  sequences  is  itself 
averageable,  and  has  for  its  value  the  sum  of  their  respective  values. 

Let  the  two  sequences 

I  Sq,  Si,   S2,    '  '  '    Sn,    '  '  ' 
I    to,    ti,    /21    •  •  ■     tn,    •  •  • 


l8  UNIVERSITY   OF   MISSOURI    STUDIES 

have  h,  h  and  nti,  mi  as  their  respective  limit-values,  and  5  and 
/  as  their  respective  sums.     Then  we  have: 


/  = 


tL  L    gi«  +  g2(i)    J- 


We  wish  to  show  that  the  sequence 

So  ~t~  ^Oi   Si   -{-  ti,    '  '  •    Sn   -T   tni    '  '  ' 

is  averageable,  and  has  for  its  value  s  -\-  t.  We  observe  that 
the  only  limit-values  for  the  sum-sequence  are  h  +  Wi,  l\  +  m2, 
U  +  Wi,  /a  +  W2.  Let  us  call  7^,i(w)  the  number  of  the  (5^  +  /,,) 
which  are  near  the  limit-value  /,•  +  rrij.  Then  we  have  to 
consider:* 

Fn(w)(/i  +  Wi)+/^12(«)(/l  +  W2)+F2l(w)(/2  +  Wi) 

+/^22(w)(/2  +  ^2) 

/^ll(«)  +  Fii{n)  +  7^2  l(w)  +  /^22(«) 

It  is  clear,  however,  that 

Fx,{i)  +  Fn{i)  =  /iW  +  ^1  U  ^nW  +  ^2i(^')  =  giC^)  +  d, 
F2,{i)  +  F22(i)  =  f2(i)  +  C2 11  Fi2{i)  +  FasC^)  =  g2(^)  +  ^2 

where  Ci,  C2,  di,  d^,  are  constants  independent  of  i.  We  ac- 
cordingly obtain: 

Fn{n){li  +  Wi)  -f  Fn(n){h  +  W2)  +  F^Mih  +  Wi) 

+  F22in){h  +  ^^2) 

^ii(w)  +  i^i2(w)  -I-  F2i{n)  +  F22(n) 

r  [Fii{n)  +  Fi2(n)]h+[F2i{n)  +  F22(n)]h 

_  y    i  +[Fll(w)  +  /^2l(w)]mi+[Fi2(«)  +  F22(«)]w2 


.  I  Fn{n)  +  7^i2(w)  +  F2i(w)  +  /^22(«) 


*  We  have  defined  averageability  for  sequences  with  only  two  limit  values. 
The  extension  to  sequences  with  any  finite  number  of  limit-values  is  obvious 
(see  page  19). 


=  L 


DEFINITION   OF    SUM    OF   A    DIVERGENT    SERIES  1 9 

[Mn)-\-c,]h+[Mn)+c.]h 

igi{n)-]-di]nii+  [g2(n)-\-d2]in2 


/l(w)+Ci+/2(w)+C2 


+  1. 


giin)-\-di-\-g2{n)-\-d2 

=  s  -\-  L 


^j    r/i(w)/i  +f2(n)l2~\       J    V gi{7i)mi  +g2(n)m2l 
ntL  L    fM  +h{n)    yHX     gr{n)  +  g2{n)      J 

Thus  it  is  seen  that  the  requirements  *  of  page  2  are  satisfied 
by  our  definition.  The  extension  of  the  definition  to  the  case  of 
sequences  with  any  finite  number  of  limit  values  is  obvious. 
Definition:  A  segiience  having  k  limit  values,  /i,  /?,  •  •  •  h,  shall 
be  called  averageahle,  and  have  s  for  its  value,  if 

■  n=i 
^fn{i)ln 


Hfnii) 

It  can  be  easily  verified  that  Theorem  i  applies  to  this  extended 
definition. 

But  we  can  generalize  the  notion  of  averageability  even  to 
cases  where  the  sequence  has  an  infinite  number  of  limit-values. 
Let  us  consider  a  reducible  sequence,  and  let  us  write: 

(E)  =  (£(0))  =  So,       si,       St,       ■••  Sn, 

(£(2))  ^  /^(2)^    ;^(2)^    ;^(2)^    . . .  i^m^    . . . 


where  the  sequence   {E'^'^)  consists  of  the  limit  values  of  the 
sequence  (£^'~'0.     Since  the  sequence  is  assumed  to  be  reducible, 
there  exists  a  k  such  that  (£(*+i))  =  o.     Then  (£)  is  reducible  of 
order  k,  and  (£^^^)  has  only  a  finite  number  of  elements. 
*  Requirement  (v)  is  satisfied  by  each  definition  considered. 


20  UNIVERSITY   OF   MISSOURI    STUDIES 

Let  US  assume  that  our  sequence  is  reducible  of  order  k,  and 
that  (£(*>)  has  for  its  elements  lo^''\  //^\  •  •  •  Ip'^'^K  If  now  we 
choose  e  sufficiently  small,  all  but  a  finite  number  of  the  /i^*~^^ 
will  fall  in  the  intervals  |  /<(*-!>  -  Ip^^^  \  <  e,  p  =  o,  i,  2,  ■  ■  ■  p. 
Suppose  that  the  finite  number  of  /.-^'^"^^  which  do  not  fall  in 
any  of  these  intervals  is  pi,  and  call  them,  w/'^"'^  m2^''~^\ 
•  •  •  mp^^''~'^K  We  can  choose  ei  <  e,  so  small  that  only  a  finite 
number,  p2,  of  the  li^^~^^  do  not  fall  in  any  of  the  intervals 
above,  or  in  the  intervals  |  /,^''~2)  _  wzp(^— 1)  |  <  gj,  ^  =  1,2,  •  •  •  pi. 
Call  this  finite  set  of  limit  points  mi^''~^\  •  •  •  mp^'-^~^\  We  can 
repeat  this  process  until  we  reach  the  sequence  (-E),  which  will 
have  only  a  finite  number  of  elements  outside  of  all  the  intervals 
considered. 

Definition:  A  reducible  sequence  shall  be  called  averageable, 
unth  s  for  its  sum,  provided* 


5  =  LL 

e=0  H=K 


j  =  l        i  =  l 


f-  =  L  F(e) 

6=0 


j=k  i=p^_j+i 

2:     Z  fi''\n,  e) 
j=i    i=i 

exists. 

In  this  general  definition  it  is  convenient  to  distinguish  between 
different  kinds  of  limit  points.  Let  us  suppose  that  /,(«,  e)  cor- 
responds to  the  limit  point  w,-,  and  let  us  assume  that  the  fol- 
lowing limit 

_  J  /.(w,  e) 

"^"  Z     Z    Mn,  e) 

j=i    1=1 

exists  for  every  i.  We  shall  call  w,-  a  iveak  or  a  strong  limit 
point  according  as  a,-  is  or  is  not  equal  to  zero.  We  may  then 
state  the  following  proposition: 

Theorem  2 :  A  reducible  averageable  sequence  with  a  finite  number 
of  strong  limit  points  is  averageable  independent  of  e. 

*  We  have  put  Wi^*^  =  /i^*^  for  the  sake  of  uniformity. 


DEFINITION    OF    SUM    OF   A   DIVERGENT    SERIES 


21 


For  simplicity  consider  the  case  where  the  reducibility  is  of 
order  2.  The  strong  limit  points  are  then  either  of  the  first 
or  of  the  second  order.  There  is  only  a  finite  number  of  strong 
limit  points  of  order  2,  and  a  finite  number  of  strong  limit 
points  of  order  i.  Let  m  be  the  total  number  of  strong  limit 
points.     Since  for  the  remaining  limit  points  a,-  =  o,  we  have 


F(e)  =  L 


fi{n,  e)h  -\-  fijn,  e)h  +  •  •  •  +/r«(w,  e)l„ 
H  fi{n,  e) 


If  we  now  choose  e'  <  e,  the  values  of  the  coefficients  of  the 
strong  limit  points  are  unaffected.  Hence  F{e')  =  F(e),  and 
our  theorem  is  proved. 

Theorem  3:  A  reducible  averageahle  sequence  with  a  finite 
number  of  strong  limit  points  is  Cesdro-summable  of  order  i ;  and 
the  two  values  obtained  are  equal. 

We  lay  off  ei  intervals  about  the  limit  points  of  order  k  —  i-\-i, 
(i  =  I,  2,  •  •  •  ^)  as  on  page  20,  and  we  thus  have  for  n  >  N, 
if  e  is  the  largest  of  the  ei, 

I  h  -  s/  \  <  e,  i  =^  I,  2,  ■  • '  fi{n,  e) 

I  h  -  Si"  I  <  e,  i  =  1,2,  •  •  •  fiin,  e) 


where  5,-^''  are  those  Si 
which  fall  in  the  e-\n- 
terval  about  /,-. 


I  Ip-Si^p^  I  <  e,  i  =  1,2,  ■  ■  ■  fp(n,  e)  _ 
We  have  accordingly: 

1  (fih  -\-f2h  +  •  •  •  +/p/p)  -   [(si'  +  •  •  •  +  Sf/)  +  . 


Since 

(5/  +  5/  + 


+  Sf,)   +    •  •  •    +  (5i(-)  +    .  •  •    +  Sf/^>^) 
=   Sm+1   "T  Sm+2   "T 


-\-fp)e. 


+  s„ 


where   q  =  /i  +  /o  + 
have : 


+  /p,    and   m   is  sufficiently   large,  we 


22  UNIVERSITY   OF   MISSOURI    STUDIES 

flh    +  f2h    +    •  •  •    +  fjJp  S„,  +  i    +   Sm+2    +    •  •  •     +   Sm+g 


<  e. 


Hence 


2  J 

provided  either  limit  exists.  By  Theorem  2,  the  left-hand 
limit  exists  independently  of  e;  accordingly  the  right-hand  limit 
exists;  that  is,  the  given  sequence  is  summable  (Ci). 

In  practice,  the  following  proposition,  a  corollary  of  the  theorem 
just  proved,  will  be  found  useful: 

Corollary:  If  for  some  positive  integer  k,  and  for  every  positive 
integer  i  <  k,  the  sequence  Si,  Si+k,  Si+2k,  •  •  •  converges,  then  the 
sequence  51,^2,  •  •  •  is  summable  (Ci). 

Let  us  take  as  an  example  the  sequence 


(•+1).  '■ 


Si  —  i  log  (  I  +  T  I ,     i  odd 
=  0,  i  even 

to  which  it  is  not  easy  to  apply  the  formula 

J      5i    -f   52    +    •  •  •    +   -^n 

n=cc  n 

We  see,  however,  that  the  two  sequences 

51,  S],    •  • 

52,  Si,    •  ■ 

converge;  hence  the  given  sequence  is  summable  (Ci). 


§  4-    PRODUCT   DEFINITIONS 

In  dealing  directly  with  sequences,  the  Cauchy-product*  of 
two  series  does  not  appear  to  be  entirely  natural.  Even  in 
the  case  of  convergent  sequences,  a  more  natural  definition  of 
product  is  close  to  hand.  In  fact,  if  s  and  t  are  the  respective 
sums  of  two  convergent  sequences, 

•^0,   Si,   52,    •  •  •   Sn,     '  '  ' 
^0)    ilt    ht     '  '  '    in,     •  •  •  > 

then   it   follows   from    a   fundamental    theorem   of  limits   that 

1j  sJh  =  St. 

n=oo 

We  are  accordingly  ledf  to  propose  the  following 
Definition :  The  natural- product  of  two  sequences, 

So,    Si,    52,    •  •  •  Sji,     •  '  •  J       to,    ti,     '  '  '   tn,     '  '  't 

is  the  sequence:  Soto,  Siti,  •  •  •  Sntn,  *  •  •• 

We  may  then  state  the  obvious  proposition: 

Theorem:  The  natural-product  of  two  convergent  sequences, 
whose  values  are  s  and  t  respectively,  is  itself  convergent;  and  its 
value  is  st. 

If  we  compare  this  theorem  with  the  corresponding  theorem  J 
for  the  Cauchy-product,  it  will  be  seen  at  once  that  the  natural- 
product  is  of  superior  value  to  the  Cauchy-product,  in  the  case 
of  convergent  sequences  of  constant  terms.  In  the  case  of 
sequences  which  are  not  convergent,  however,  the  natural- 
product  can  play  no  part.     For  consider  the  simple  example, 

*  Sec  page  5. 

t  Baire:  Cours  D' analyse,  t.  i. 

X  Theorem  b,  page  6. 

23 


24 


UNIVERSITY   OF    MISSOURI    STUDIES 


S  =   I,    O,    I,    O,    • •• 

/  =   I,    O,    I,    O,    ••  • 

W  =   I,    O,    I,    O,    •••, 

where  the  sequence  whose  value  is  w  is  the  natural-product  of 
the  two  sequences  whose  values  are  5  and  t  respectively.  Here 
s  =  t  =  w  =  ^,  and  accordingly  w  4=  st.  We  ^re  consequently 
led  to  generalize  the  definition  for  the  product  of  two  sequences. 
Let  us  consider  again  the  two  sequences 

I  So,  Si,  52,    '  '  '   Sn,    '  '  ' 

L  to,    ti,    t2,    •  •  '   tn,     •  •  • 


and  let  us  form  the  array: 


Soto,  \ 

Sotl, 
Slti, 

Sot2, 
Slt2, 
Siti, 

Sotn, 
Sltn, 
Sitn, 

SnPn, 

Sito, 

S2to, 

S2tl, 

SjO' 

Sjl, 

Snt2, 

•     •     • 

Definition:  The  sequence  formed  by  following  the  successive 
lines  which  form  squares  with  the  boundaries  of  the  array,  i.  e., 

•^0^0,  •^0^1,  Siti,  Sito',  SOfti,   Sit2,  ^2^2,  -^2^1,  •^2^01    '  '  '» 

shall  be  called  the  square-product  of  the  two  sequences. 

We  shall  now  prove  the  following  theorem : 

Theorem  4:  The  square-product  of  two  averageable  sequences 
is  averageable,  and  its  value  is  equal  to  the  product  of  their  values. 

Let  the  given  sequences  be 


DEFINITION   OF   SUM   OF   A  DIVERGENT   SERIES 


25 


5  —  So,  Si, 

t   —  to,    t\. 


we  wish  to  prove  that  the  sequence 

^O^O!   •^0^1)    •^1^1)    -^1^0!    -^0^2)    -^1^2)    -^2^21    •^2^1.    -^2^01     "  '  ' 

is  averageable,  and  that  its  value  is  st.  We  shall  assume*  that 
the  sequence  (5)  has  the  two  limit-values  h,  h,  and  that  the 
sequence  (t)  has  the  two  limit-values  nti,  nh.  The  only  limit- 
values  of  the  product  sequence  are  then:  hnii,  /1W2,  hmi  and  /2W2. 
We  are  given 

+  /2(w)/2 


^  J    \ Mn)h  +  f2(n)hl 
nti  I  Mn)-hMn)    J 


«=«  L  giw 

and  we  wish  to  consider: 

n=co  L 


+  /2(W) 
+  ^2(w)/2 


+  gM 


I 


{n)hmi  +  Fn{n)li'm2  +  Fii{n)h'mi  -\-  F2i{n)l2m2 
Fnin)  +  Fuin)  -f-  i^2i(«)  +  i^22(«) 


]■ 


where  Fij{n)  is  the  number  of  elements  of  the  product  sequence 
near  Umj.  If  we  pick  n  elements  from  the  product  sequence, 
we  observe: 

Fii{n)  =  fi(n)gi{n)  +  ^n  1  f  F^iin)  =  /2(w)gi(«)  +  ^21 


hi  I  r  F2i{n)  = 
h2  J  I  F22{n)  = 


Fn{n)  =  fi{n)g2{n)  -f  ^12  J  I  -F22(w)  =  f2{n)g2{n)  -f  ^22, 
where  kij  are  constants  independent  of  n.     We  have,  accordingly, 


r^n 


{n)limi  -f  Fn(?i)lini2  +  F2i{n)l2fni  -f  F22(n)kfn2 
Fn(n)  +  Fn{n)  +  F2i{n)  +  F22(n)  . 


[/i(w)gi(w)  +  kn]hmi  -f  [fi(n)g2{n)  +  kn]hni2 

+  [f2(n)gi(n)  +  k2i]hmi  -\-  [f2{n)goin)  -f  k22]hm2 
fiin)gi{n)  +  kn  +/i(w)g2(»)  +  ^12  +/2(w)gi(«) 

+  ^21  +f2{n)g2{n)  +  ^22 


The  proof  for  the  general  case  is  precisely  similar. 


26  UNIVERSITY   OF  MISSOURI    STUDIES 


=  L 


~fi(n)gi{n)hmi  +/i(«)g2(w)/im2  +f2{n)gi(n)l2tni 

+  f2{n)Z2{n)hm2 


.»=«  L     Mn)gi{n)  +fi{n)g2{n)  +f2in)gi{n)  -]r  h{n)g2{n) 
J    r  [hWi  -^Ji{7i)h]  [gx{n)mi  +  g2(w)w2]  1  ^ 

For  example,  the  square-product  of  the  sequences 

f  5  =  I,  o,   I,  o,  •  •  • 

1/  =  I,  o,  I,  o,  •  •  •, 
is 

w^=  i;  o,  o,  o;  i,  o,  i,  o,  i ;  o,  o, o,  o,  o,  o,  o;  1,0,1,0,1,0,1,0;  •  •  •. 
If  we  choose  m  terms  of  this  sequence,  and  let  (2«)^  be  the  largest 
square  of  an  even  integer  less  than  or  equal  to  w,  so  that 

m  =  {27iY  -\-  k,     o  <  ^  <  8«  +  4, 
we  get: 

T   f  [i+3H h(2«-i)]i+[w-(iH l-2w-i)]-o 

»=«  m      „=«,  4^2  -f  k 
_  1 

Thus  it  is  verified  that  w  =  s  •  t. 

Although  it  is  true  that  the  natural-product  is  better  adapted 
to  convergent  sequences  than  the  Cauchy-product,  and  that  the 
square-product  is  better  suited  for  averageable  sequences,  it  must 
be  remembered  that  in  analysis  the  things  that  arise  frequently 
are  not  sequences  of  constant  terms,  but  rather  series  of  variable 
terms,  notably  power  series.  In  the  case  of  power  sei'ies,  the 
Cauchy-product  is  certainly  more  valuable;  for  if  we  multiply 
two  such  series  according  to  the  Cauchy  scheme,  we  obtain  the 
same  result  which  is  given  by  multiplying  the  two  series  as  if 
they  were  polynomials,  thus: 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES  27 

{U(x)    =   «o  +  UiX  +  U2X'^  4-  UsX^  +    •  •  •    +  Wn^"  -f-    •  .  . 
V{x)    =    Vo  +  ViX    +  ViX"^    +  VzX^    -f-    .  .  .    4.  VnX''    +    •  •  • 

ze'(A;)=w(x)-f(x)  =  z^oyo+(«oi'i+Wii'o)x+(wo?;2+WiZ'i+M2Z'o):'c^H . 

Furthermore,  to  this  symbolic  advantage  is  added  the  theoretical 
one  which  is  contained  in  the  following  theorem,  due  to  Ces^ro,* 
which  is  a  generalization  of  Theorem  b. 

Theorem  (j):  The  Cauchy-product  of  two  Cesdro-summable 
series,  of  orders  p  and  q,  and  of  values  s  and  t  respectively,  is  itself 
Cesdro-summable  of  order  at  most  p  -{■  2  -\-  i,  and  its  value  is  st. 

In  certain  special  cases,  we  can  slightly  improve  upon  the 
results  of  Ces^ro's  theorem.  Thus,  if  two  series  are  convergent 
(i.  e.,  summable  of  order  o),  their  product  must  be  summable 
of  order  at  most  i.  If,  however,  one  of  these  series  converges 
absolutely,  then  the  product-series  is  convergent,!  as  has  already 
been  stated.  J  Similarly,  the  Cauchy-product  of  two  Cesaro- 
summable  series,  one  of  order  r,  the  other  convergent,  is  sum- 
mable (C+i) ;  if  the  convergent  series  happens  to  be  absolutely 
convergent,  however,  the  product  can  be  shown  to  be  summable 

(Cr). 

Theorem  5 :  The  Cauchy-product  of  a  Cesdro-summable  series 
of  order  r  by  an  absolutely  convergent  series,  is  itself  Cesdro-sum- 
mable of  order  r. 

Let 

^n    =   Wo  +  Wl   +    •  •  •    +  W„, 

tn    ==   Vq  -\-  Vi   -\-    "  '    -\-  Vn, 

Wn    =    thVn  +  UiVn-\   +    *  "  '    +  UuVq, 

3'n   =   Wo  +  Wl  +    •  •  •    +  W„. 


*  Ces^ro:  Bull,  des  Sciences  math.,  t.  XIV,  1890. 
t  Mertens,  Journal  de  Crelle,  t.  79,  p.  182. 
J  P.  5,  supra. 


28 


Yn  =  y 


UNIVERSITY  OF   MISSOURI    STUDIES 

r{r  -\-  i)  •  •  ■  {r  -\-  n  -  i)    ,        r{r  -\-  \)-  •  •  {r  -\-  n  -  2) 


111 


+  yi 


{n-  i)\ 


r(r  +  i) 
+  •••  -\- yn-2 — —^ — -\-yn-i'r+yn, 

r(r  +  i)  •••  (r  + w  -  i)  r(r  +  i)  -  ■  ■  (r  +  n  -  2) 


+    •  •  •    +   /n- 


,       ,       r(,r  +  1)  •  •  •  {r  +  n  -  i) 


(«  -  i)! 
r{r  +  i) 


2! 


+  /„_!•;'  +  /„, 


(r,  w) 


We  assume: 

\jSn    =   S,         Ij    [|Mo|   +   |Wl|   +    •  •  •    +   Vln\   ]    =   A, 


T 


n=^  (r  +  i)  •  •  •  (r  4-  w) 


=  i, 


and  we  wish  to  prove : 


Yn 


t=«  (r  -\-  i)  ■ '  ■  (r  -{-  n) 


=  s-t. 


Proof: 
Lemma:  If 


.=00  (r-hi)  •■•  (r-\-n) 


n\ 


=  t,     then     Ij 


T    —  T 

(r  +  i)  •  •  •  (r  +  w) 


o, 


For 


^  =  I,  2,  •••  p. 


-1  n 


T 


(;-  +  i)  •  •  •  (r  +  w)       (r  +  i)  •  •  •  (r  +  w) 


w! 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES 

T  T 

■'■  n  J-  n—p 


29 


n=8 


(r -{- i)  ■  ■  ■  (r -\- n)       (r -\- i)  -  ■  -  {r +  n  -  p) 


n 


{n-p)\ 
{r-\-i)  "■  {r-\-n  -  p) 


(n  -  p)\ 


(r  +  i)  •••(/'  +  w) 


T 


T 


{r-\-i)-  •  ■(r-\-n)       (r-\-i)  •  •  •(r-{-n—  p) 
n\  {n  —  p)\ 

n{n  —  i)  '  ■  •  {n  —  p-\-i) 
(j  +  n)- {r-\-  n  —  i)-  -  -  {r  -\-n  —  p-\-  \) 


t  —  t  -  I  =  o, 


Now 


y,,  =  uovo  +  (tiovi  +  wiz'o)  +  (U0V2  4-  wiz/i  4-  U2V0)  4-  •  •  • 

+  {UoVn  +  UiVn-l  +    •  •  •    +  tln~lVl  +  UnVo) 
=    Uo{Vo  +  Vi-{'   •  •  •    -\-Vn)  +  UliVo  +  Vi-\-  ■■  ■   -\-  Vn-l)  +    •  •  ' 

+  Un-l{Vo  +  Vl)  +  UnVo, 
yn    =   Wo^n  +  Ml/n-1  +    '  '  "    4"  «n-l^l  4"   l^Jo- 

r{r  +  i)  •  •  •  {r  -^  n  —  i) 


Yn    =    UotQ 


+  (i^oi^i  +  Uito) 


r{r  -\-  i)  •  •  •  (r  -j-  n  —  2) 
(n-i)\ 


r(r -\- i) 

4-   •  •  •    4-   {Uotn-2  +    •  •  •    4-  tin-2to) ^"j 

4-(z<o/„_lH \-Un-lto)r  +  (Uotn-] ["Wn^o). 

Yn   =    UoTn  +  UlTn-l  4"    •  •  •    4"  Un-lTi  4"  Wn^^O, 
Yin    =    UoTin  4-  UlTin-l   4"    '  *  '    4"   thn-lTi  4"  U^nTo, 


30 

Let 
R  = 


UNIVERSITY  OF   MISSOURI   STUDIES 


*7l 


(2W) 


(npTin  +  UiT2n~\  +    •  •  •    +  thn-lTi  +  thnTp) 

{r  +  i)(r  +  2)  •  •  •  (r  +  2w) 
(2«) ! 

Tn 

—    (tlo  +  Wl  +    •  •  •    +  Mn) 


(r  +   l)    •  •  •    (r  +  «) 


R  <     U<o 


7^2  n 


-t   n 


+ 


I  l«g+i 


(r+i,2w)        {r+i,n) 

+  •••  +  !« 

7^2n— g-l 


-/  2n-l 


-»  n 


{r-\-i,2n)       (r+i,w) 


(r+i,2w)      (r  + 
T 


^11 
i,w)|  J 


4-      |Wn+l| 


<  Ml   +   Ms  +   |Wg+ll 


(r  +  I,  2«)       (r  4-  ii  w) 

+    Wn\ 

r„_i 


+ 


-^   n 


(r  +  I,2w)         (r  + 


r  +  I,  2W 

7^2n-g-l 


+    •  •  •    +     W2n| 


+ 


M1 

I,  2W)  I  J 


f+I,2W  —  g—  l|         |(r  +  I,  w) 


+    •  •  •    +  \Un\ 


T 


(r  +  i,  w) 


+ 


{r  + 


^11 
I,«)1J' 


where  Mi,  M2  and  M3  stand  respectively  for  the  expressions  in 
the  first,  second  and  third  brackets  above. 


since 


Also 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES 


31 


!      B 
<  —  for  all  m. 
2 


(r  +  I,  m) 

M,  <   {|«„+i|  +  •••  +  \lhn\}B. 


Now  as  to  Mu 

■i  2n— p  ^  n 


< 


In 


T 


{r-\-i,2n)      {r-\-i,n)        |(r+i,2w)       {r-\-i,n) 


+ 


< 


(r+i,2w)      (r+i,2w) 
5  5 


;     p  =0,1,2, 


2{A  +  By  2{A  +B) 


,iin>N; 


'.R  <  {\uo\  +  |mi|  +  •••  +  |Mg|}  jqr^ 

+  {\u,+i\  +  •  •  •  +  \un\  +  •  •  •  +  \u2n\\B,  iin>  N. 
Now  choose  q  so  large,  that 

\u,^i\+  "■  +\u2n\<j-^,     q>  Q  (or  all  n. 

Moreover,  |wo|  +  •  •  •  +  |w,|  <  ^  for  all  q. 


eA  -\-  eB 
:.  R  <     .    ■    ^    =  e. 


Thus 


Similarly 


A-\-B 

F2„ 


p     y2n+l 

V^{r^  i,2w+  i) 


=  s-t. 


=  s-^. 


The  theorem  is  now  proved. 

In  the  case  of  power  series,  then,  both  the  symbolic  advan- 
tage and  the  theoretical  importance  of  Theorems  j  and  5  lead 


32  UNIVERSITY   OF   MISSOURI    STUDIES 

naturally  to  the  Cauchy-product.  This  advantage  does  not  ap- 
pear, however,  in  case  of  sequences  which  do  not  correspond 
to  power  series, — for  example,  in  Fourier's  series;  in  this  case, 
the  square-product  may  be  of  greater  service  than  the  Cauchy- 
product,  We  should  observe,  however,  that  while  the  square- 
product  may  justly  replace  the  Cauchy  definition  of  multipli- 
cation, in  certain  cases;  the  definition  of  averageability  has  the 
disadvantage  of  presupposing  the  knowledge  of  the  limit- values; 
and  these  are  not  always  easy  to  determine  even  in  the  case  of 
sequences  of  constant  terms. 


§  5-    ON  CERTAIN  POSSIBLE  DEFINITIONS  OF  SUMMABILITY 

Cauchy  has  proved*  the  following  theorem,  which  we  shall 
show  is  equivalent  to  Theorem  c. 
Theorem  k:  If  Un>  o  and 

L  ^-1^  =  /,     then     L  «„!/«  =  /. 


n=oo     "-n 

Let 


=    tn+X,        Uq    —    I, 
Un 

then 

Un    =    titi    •  •  •    tn- 

Accordingly,  whenever 

n=«o 

then 

L  (/1/2  •  •  •  Lf"  =  t, 

«=« 

provided  tn  >  o;  and  the  last  equation  may  be  written 

-r      (  log  h  +  log  t2   -\-    •  ■  '    +  log  tn  \ 

And  if  we  finally  write  log  /„  =  5„,  we  obtain  the  result  that 

L =  ^ 

n=ao  W 

whenever 

J_J   Sn    —    S. 

7l=co 

This  statement  is,  however,  precisely  Theorem  c.  We  see 
accordingly  that  Theorems  c  and  K  are  equivalent,  by  means  of 
the  substitution 

*  Cours  d' Analyse:  Oeuvres  de  Cauchy  (2°  serie),  Vol.  3,  pt.  3. 

33 


34  UNIVERSITY   OF   MISSOURI    STUDIES 

Un 

Let  us  make  the  further  substitution  5„  =  rn(Pn,  and  observe 
that  the  variables  5  „  and  r„  on  each  side  of  this  equation  approach 
the  same  limit,  provided 

Ju    <Pn    =    I- 

We  may  accordingly  replace  Theorem  c,  which  we  have  just 
obtained  again,  by  the  following  theorem: 
Theorem  6:  If 

Tj  Tn  =  r,     and    L/  cpn  =  i, 
then 

n=^\_  n  J 

If  we  put  a  further  restriction  on  the  sequence  <^n  we  can 
broaden  the  requirement  on  the  sequence  r„.  In  fact,  we  may 
say: 

Theorem  7:  // 

X   ri  +  r2-\-  '  "  -]-  Yn 

Li =  r, 

n=»  n 

and 

monotonically*  then 

*  That  the  theorem  is  not  true  in  general,  when 

^      <Pn   =   I 
n^  CO 

not  monotonically,  follows  from  the  example: 

r„  =  (-  i)"+MogM,     v^n  =  I  +  (-  1)"+' -^,     «+i,     fi=i. 

log  n 

Here 

T      ^1  +   •  •  •   +  ''"         ^  T 

-Li     =0,        1j     ^„  =  I 

«=  CO  M  n=  cc 


wo/  monotonically; 

I. 


L,       yi?"!   +    •  •  •    +  'P«?'n 


»— -  00  n 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES  35 

LtpiU  +  (p2r2  4-  •  •  •  +  <Pnrn 
_  y 

n=oc  W 

The  proof  of  this  theorem  follows  at  once  from  the  following 
theorem  due  to  Hardy;*  for  a  proof  of  which  see  page  85. 
Theorem  l:     //  2c„  is  a  divergent  series  of  positive  terms,  then 

T     ^0^0  +  ^1^1   +    •  •  •    +  CnSn   _    J      ^0  +  ^1   +    •  •  •    -\-  Sn 
«=«,  W  +   I  „=„  W  +   I  ' 

provided  that  the  second  limit  exists  and  either 

(a)  Cn  steadily  decreases, 

(b)  Cn  steadily  increases,  subject  to  the  condition 

nc„  <  (co  +  ci  +  •  "  -\-  Cn)K, 

where  K  is  a  fixed  number. 

We  shall  now  show  that  Theorem  7  is  a  special  case  of  Theorem 
L.     In  the  first  place,  since 

Jj  (pn  =   I, 

it  follows  from  Theorem  c  that 

L<Pl  -{-  <P2  -\-   •  •  '   -\-  <Pn 
=    I, 

n— 00  n 

and  accordingly, 

J    <Piri  +  ^2^2  +  •  •  •  +  <Pnrn 

n=ao  n 

(Piri   +    (^2^2   +    •  •  •    +    <Pnrn       Vl    +    9^2   +    '  '  "    +    <Pn 


^1   +   ^2  +    •  •  •    +   <^n 

<Piri  +  (pzrz  +  •  •  •  +  <Pnrn 


<Pl   -\-    <P2   -{-'''    -\-   fn 

We   may  now   apply  Theorem   L  directly,   by   identifying   ^„ 

*  Quarterly  Journal,  Vol.  38  (1907),  p.  269.  Hardy  proves  a  more  general 
theorem  of  which  this  is  a  special  case;  the  first  part  of  the  general  theorem  has 
been  first  proved,  however,  by  Ces^ro,  as  Hardy  himself  states.  See  Cesiro: 
Bull,  des  Sciences  math.  (2),  t.  13,  1889,  p.  51. 


36  UNIVERSITY   OF   MISSOURI   STUDIES 

with  Cn.  If  Vn  decreases  monotonically,  the  condition  of  the 
first  part  of  Theorem  L  is  fulfilled;  if  v'n  increases  monotonically, 
we  have: 

^1   +   ^2  +    •  •  •    +   <Pn    >   W^l, 

or 

so  that 

^    T-^  (^1   +    <P2   +    •  •  •    +    ^r) 
ipn<  IS.  -  , 

n 

which  is  precisely  the  second  requirement  of  Theorem  L.  Hence 
the  truth  of  Theorem  7  is  established. 

We  can  deduce  an  interesting  consequence  from  Theorem  7, 
and  say,  in  the  language  of  §  4, 

Theorem  8 :  The  natural  product  of  two  sequences,  one  of  ivhich 
is  summahle  of  order  i,  the  other  monotonically  convergent,  is 
summahle  of  order  i ;  and  the  value  of  the  product  sequence  is 
equal  to  the  product  of  the  values  of  the  two  given  sequences. 

Let  Sn  and  /«  be  the  two  given  sequences, 

L-^l   +  -^2   ~t~    •  •  •    -^  Sn  -|- 

—    S,         l^tn    —    t, 
71=00  ''  n=a) 

monotonically.  We  first  suppose  that  /  =]=  o,  and  form  the 
sequence  tjt,  so  that 

Lf  n 
«=«  t 

monotonically.     Accordingly,  by  Theorem  7, 
^1     ,         ^2    ,  ,         in 

-L =  5 


or 


lu =  St. 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  37 

If  /  =  o,  we  form  the  sequence  i  +  /„,  so  that 

L  (i  +  /  J  =  I 

n=oo 

monotonically ;  consequently,  by  Theorem  7, 

^   ^     J^   ^1(1    +   h)    +   52(1    +   /2)    +    •  •  •     +   ^n(l    +   tn) 

«=«>  n 

^    J     -^1  +  •?2  +    •  •  •    +  Sn  -|-     ^1/1   +  52^2  +    •  •  •    +  Sntn 

n=co  H  n=tx>  W 

and  accordingly, 

lu   =   O. 

n=oo  n 

Let  us  now  return  to  Theorem  6,  and  base  upon  it  the  following 
definition: 

Definition :  The  sequence  shall  be  said  to  be  <p-suinmable,  and  to 
have  the  value  s,  provided 

LSl<Pl   -\-  S2(P2  +    •  •  •    +  Sn<pn 
n=ca  n 

L  <pre  =  I. 

It  is  natural  to  ask  for  the  relation  between  <p-summability 
and  Ces^ro-summability,  In  general  it  will  be  possible  to  find  a 
sequence  <pn  which  will  give  a  more  general  definition  than  that 
of  Ces^ro-summability  of  order  i.  We  can  however  restrict  the 
sequence  ^„  so  as  to  make  the  two  definitions  equivalent ;  and  we 
may  state  the  following  theorem: 

Theorem  9:  If 

monotonically,  then  whenever  either  of  the  two  definitions — ^- 
summability  or  Cesd,ro-summability  of  order  i — gives  a  value  to  a 
given  sequence,  so  will  the  other,  and  the  two  values  will  be  the  same. 


T^ 


38  UNIVERSITY   OF   MISSOURI   STUDIES 

If  we  choose  any  specific  sequence  ^„,  subject  to  the  condition 

n=«> 

monotonically,  then  it  follows  at  once  from  Theorem  7  that  if 
a  sequence  is  summable  of  order  i,  it  is  also  ^-summable  for 
the  particular  ^„.  Let  us  now  suppose,  conversely,  that  the 
sequence  5i,  52,  •  •  •  5„,  •  •  •  is  ^-summable  for  ^„,  i.  e., 

Li =  s. 

This  amounts  to  saying  that  the  sequence  (5„<p„)  is  Ces^ro- 
summable  of  order  i.     Let  us  now  apply  Theorem  7,  making 

r„  =  Sn^Pn,  and  (p„  =  i/lpn-     Since 

monotonically,  then 
monotonically,  and 

_   J     pl^l«pl  +  S2(P2<P2  +   •  •  •   +  ^n^nlpn"]  _    j     Si  -]-  S2  -\-  '  •  •   -\-  Sn 

«=1  L  n  J     „=„  n 

i.  e.,  the  given  sequence  is  Ces^ro-summable  of  order  i. 
If  we  assume  that 

L  «Pn    =    I 

non-monotonically,  then  Theorem  7  may  no  longer  apply,  as 
is  shown  by  the  following  example: 

r  (fi  =  1 


Si  =  (-  i)'+Mogi^  I 

so  that 


^'  =  '  +  (-"'"i^-'=^'3' 


Si<pi  =  o 

Siifi  =  I  4-  (-  1)'+^  log*  =  I  -\-  Si,  i  =  2,  3, 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES 


39 


Now 

T     ^1  +  ^2  +   •  •  •   +  -^n  _   J     log  I   -  log  2  + 


log  n 


=  o' 


and 

71=00 

non-monotonically. 

If  Theorem  7  were  true,  ipn  non-monotonic,  we  should  have 


JL =  o; 


whereas, 


J      Siipi  +    •••    -\-Sr.ipn   _    y      log  I   +  (l   +  ^2)  +    '  '  '    +  (l    +  Sn) 

=  Li 1-  JL =  I. 

n=a)        ^  n=oo  W 

Returning  now  to  the  monotonic  ^-definition,  we  observe  that 
if  we  take  <pn  —  i,  we  obtain  Cesaro-summability  of  order  i. 
Taking 

we  obtain: 

r5ilog2  +  52 log  (I  +  hY  +  ^3  log  (I  +  i)'  +  •  •  • 

+  5nl0g(l+^) 


(6)     5  =  L 


*  L  -  2  (-i)«log.=  i  L    log  0-3--^^"-^^) 

n= «  2n  i=i  *"  2  n= «       *"    \2  •  4  •  •  •     2«  / 


-    li     log  ttn''"  =  O 
2  n=  » 


Also 


L    -^''3\-i)-log.-  =  o+     3.    12^(^5^11)  =  o. 

n= »  2«  +  I   if  1     "■  ^  n=*        2M  +  I 


40 
Since 


UNIVERSITY   OF   MISSOURI   STUDIES 


.,ii'°«(-+0"= 


monotonically,  however,  it  follows  that  this  definition  is  equiva- 
lent to  Ces^ro-summability  of  order  i,  or  (what  amounts  to  the 
same  thing)  equivalent  to  Holder-summability  of  order  i.  If 
we  now  write 

5l   +  52   +    •  •  •    +  Sn 


tn 


so  that 


ntn   -    {n    -    l)tn-l    =    Sn, 

we  may  repeat  the  process  for  the  sequence  tn,  obtaining 

/l  log  2+  /slog  (I   +^)2+    •••    +/„log(^^-^j 


=  L 


5ilog2  +  (51+^2)  login h(5i+52H h^Jlog 


«  +  I 


(7)  5  =  L 


Si  log ;—   +52  log ;—  +    •  •  •    +  5„Iog 


«  4-  I 


Since  (6)  is  equivalent  to  the  Holder-summability  of  5„  of 
order  i,  it  follows  that  (7)  is  equivalent  to  Holder-summability 
of  /„  of  order  i,  i.  e.,  with  Holder-summability  of  5„  of  order  2. 

Let  us  now  return  to  our  definition  of  v'-summability,  and 
repeat  the  process  for  another  function  i/'(w),  where 


Ij  )/'(«)     =     I, 


DEFINITION   OF    SUM   OF  A   DIVERGENT   SERIES 


41 


Writing 


we  obtain 


tn    = 


ip{\)si  +  (^(2)52  +  •  •  •  +  ^{n)Sn 


n=«>  L 


+  rP{2)t2  +  ^P{n)tn 


(8) 


=  LJ 


5i  <p(i)  ^  4^{i)  +  -^  +  •  •  •  +  -^ 


+  52^(2)  \  -~  -\ H H h5rt<p(w)  — — - 


Now,  if 


then 


and 


Li  (p(n)  =   L/iACw)  =  I, 


L-r     <Pl    -{-    <P2    -\-    •  •  •    -\-    (Pn 


n=i»  n^ico 


lA  (i)ri  +  )A(2)r2  +  •  •  •  +  </'(")''« 


(9)    _ 


L 

7i=00  ^ 


«P(I)W(I)   +^+    •••    +"^ 


+ 


<P(2){ 


;A(2) 


"AW 


V'(") 


2  w    J  n 


=  I. 


Instead  of  taking 


Jjifin)  =  Iji/'(«)  =    I, 


we  shall  assume  more  generally  that  (9)  is  satisfied,  and  take  as 
our  definition, 


42 


UNIVERSITY   OF   MISSOURI   STUDIES 


(lo) 


sMi)\  "Ail)  +  -7—  +  •  •  •  + 

L  2  n 

V(2)    .  ,    iACh) 


r  =  s, 


If  v'(w)  =  i^-Cw)  =  I,  we  obtain: 


=  L 


N[.+i+ •••+^]+4^+ ••+',]+■  •+4^ 


X       r  5i  +  52         5i  +  52   +  53    ,  5i  +  52  +  5„ 

=  jL  J  5i  +  • — - —  + : h  •  •  •  H 


=  1j      where     /„  = 

«=.«  L  w  J 

which  is  Holder  summability  of  order  2. 
If 


5i  +  52  +    •  •  •    -h  Sn 


(p{n)  =  2«,     \p{n)  = 


we  obtain: 

5i2 


W  +  l' 


5=L 

»  =  00 


+  52-2 


[1.2+2.3+  '■■  '^n{n  +  i)J 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES 


43 


=  L 

n=oo 


2«  (2n  —  i) 

•^1  TT~T  +  -^2  ,     .      +    •  •  •    +  Sn 


W  +   I 


W  +  I 


2 

(w  +  l) 


W5i   +(W   -    l)52  +    •••    +^n; 

w(w  +  l) 

2l 


which  is  Ces^ro-summable  of  order  2. 
If  we  put 

(Pn   ^    I,       "An    =   W  log(    I    +  -J, 


we  obtain: 


5  =  L 


>l|log2+log^I+^)  +   •••   +log^I  +  M| 

+52{l0g(l+^)+.  .  •  +log  (i+^)}  +  .  .  . 

+  .n{l0g(l+^)} 


Si  {  (log  2   -  log  l)   +   (log  3   -  log  2)  +    •  •  • 

+  (log  (W  +   l)    -  log  «)!+•••+  5„  [log       ^       I 


71  -\-  I                       «  4-  I                                ,        W  +    I 
5i  log ;^ +  52  log +    •  •  •    +  5n  log 


which  is  (7). 

We  have  thus  seen  that  the  definitions  of  ^-summability  and 
(10)  include  some  of  the  specific  definitions  which  we  have  already 
discussed.  One  might  naturally  ask,  however,  whether  these 
general  definitions  themselves  may  be  of  any  use.  One  use 
immediately  presents  itself,  as  can  be  seen  in  the  following 
example. 


44  UNIVERSITY   OF   MISSOURI   STUDIES 

It  is  desired  to  know  whether  the  scries  given  by 
Si  =  / v",     i  =  odd 


i\og 


(-0 


=  0  ,     1  =  even 

is  summable*  according  to  Cesaro's  definition;  and  if  so,  its 
value  is  required.  To  determine  this  directly  from  Cesiro's 
definition  requires  some  manipulation.     If  we  choose,  however, 


<Pi  =  i  log 


(■n). 


we  obtain 


Siipi  +  Si(p2-\ \-Sn^n  _  J    1  +  0+  I  +  o  H [-  o  or  I 


n 


n        n 
-  or  -  +  I 

m=oo  11' 


And  since 

L  ^n  =  I 

7l=co 

monotonically,  it  follows  that 

L-^i  +  ■^2  +  •  •  •   -\r  Sn        1 
—    2- 

n^o.  n 

This  example  leads  us  to  formulate  the  following  proposition, 
which  is  of  practical  importance: 

Theorem  io:  To  test  a  given  sequence  for  CescLro-siimmahility 
of  order  I,  a7iy  cojivenient  ipn  niay  be  chosen,  provided 

n=oo 

monotonically . 

Similarly   we   may   sometimes   simplify   our   calculations   in 

testing  for  Cesaro-summability  of  order  2,   if  we  can  find  a 

suitable  (pn  and  \pn. 

*  This  example  has  been  already  considered  from  another  standpoint. 
See  p.  22. 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  45 

We  might  now  proceed  to  generalize  to  /)-functions,  and  show 
that  the  resulting  generaHzations  would  include  all  of  Ces^ro's 
and  Holder's  definitions.  And  from  what  has  preceded,  it  is 
easily  seen  that  if  we  take  all  the  />-f unctions  equal  to  unity, 
we  shall  obtain  all  of  Holder's  forms;  while  by  a  suitable  choice 
of  these  ^-functions,  all  of  the  Ces^ro-forms  might  also  be  ob- 
tained. But  though  the  process  is  quite  clearly  defined,  the 
algebraic  details  become  too  complicated  to  carry  this  work 
any  further.  The  fact,  however,  that  we  may  use,  as  a  definition 
of  summability,  the  limit  of  an  expression  in  which  the  coeflficients 
of  the  Si  are  not  specifically  named,  but  are  given  in  terms  of 
functions  satisfying  certain  conditions,  suggests  a  more  general 
view  of  summability,  which  we  shall  proceed  to  develop  in  the 
next  article. 


§  6.   DEFINITIONS   OF   EVALUABILITY 

We  have  now  considered  a  large  number  of  definitions  of 
summability.  It  is  natural  to  ask  whether  all  those  definitions 
do  not  have  some  common  properties.  Excepting  for  the  moment 
Borel's  definitions,  to  which  we  shall  return  later,  we  can  say 
that  all*  the  definitions  of  summability  which  we  have  considered 
have  the  following  properties  in  common: 

If  Uiin)  represents  the  coefficient  of  Si  in  any  of  the  expressions 
whose  limit  gives  rise  to  one  of  the  definitions  of  summability, 
then: 
(i)  L  ai(n)  =  o,  for  fixed  i, 

(ii)  L  [ai{n)  +  a^in)  +  •  •  •  +  a„(w)]  =  i, 

(iii)  ai{n)  >t  o  for  all  i  and  n. 

That  properties  (i)  and  (iii)  are  common  to  all*  of  the  definitions 
under  consideration  is  easily  verified.  We  proceed  to  show  that 
the  same  is  true  of  property  (ii).  Beginning  with  Ces^ro- 
summability  of  order  r,  we  shall  show  that  the  sum  of  the  coef- 
ficients of  the  numerator,  divided  by  the  denominator,  is  iden- 
tically equal  to  unity.     For  this  purpose  we  write: 

(i  -  :t)-('-+i)  =  (i  +  X  -1-  x2  +  jc3  •  •  •  +  jc"  +  •  •  •)(!  -  x)-\ 
Equating  the  coefficients  of  x"  on  each  side  of  this  identity,  we 
obtain : 

(r  -{-  i)(r  -\-  2)  •■■  (r  -{-  n)  r{r  +  i) 
nl -i+r  +  -^T-+--- 

r(r  -\-  i)  ■  •  •  (r  -]-  n  -  i) 


*  We  exclude  also  definition  (lo). 

t  The  equality  sign  occurs  in  the  case  of  convergence. 

46 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES  47 

SO  that: 

r(r  +  i)  (r  +  2)  •  •  •  (r  +  w  -  i)    ,  r(r  +  i) 

(r+  i)(r  +  2)  •••  (r+w)  -^• 

«! 

Turning  now  to  Holder's  definitions,  we  observe  that  for 
order  i,  the  sum  of  the  coefficients  of  the  Si  is  identically  equal 
to  unity — this  being  in  fact  a  special  case  of  the  case  just  con- 
sidered. Suppose  now  that  hi,  hi,  •  •  •//„  are  the  coefficients  of 
Holder's  definition  of  order  p,  so  that 

Ij  [^i5i  +  hiSi  +    •  •  •    +  hnSn]   =  s. 

71=00 

If  we  assume   that  hi  -\-  hi  -{-  •  •  •  +/;„  =  !    for  order  p,   we 
obtain  for  order  p  -\-  i,  putting 

^"  ~                 n 
-L    hiti  +  hi  — - —  +•••+//„ 

n=:«  L  2  n  J 


n 


hn 

n 


and  the  sum  of  the  coefficients  becomes 

r,,+^^+...+'i.i+r^'+...+^-^]+...+ 

L  2  n  J       i_  2  n  J 

=  hi  -{-  hi  -{-  •  ■  ■  -\-  h„  =  I. 

Thus  the  proof  of  (ii)  for  Holder's  definitions  is  completed  by 
mathematical  induction. 

Let  us  now  consider  formula  (7).     We  shall  show  that 

Ij  w„  =  I, 


48  UNIVERSITY   OF   MISSOURI   STUDIES 

where 


log  -  +  log  ^  +  •  •  •  +  log  ^^-3^ 
n 


«„  =  

If 

then 

Hence 

Accordingly, 


( n    n  «    X;: 


n    n 
Vn  =  - 


I     2        n  —  I       (w  —  i) ! 


Pn+l 


(-;)■ 


'rt=oD  71=0)     <' n 


Ll  nn    =    L/logi'n''"    =    I. 


Finally  since  we  have  assumed  in  the  (;c-definition  that 
Ij  ip{n)  =  I, 

n=oo 

it  follows  that 

y(l)  +  'p(2)  +    •  •  •    +  <p(n)   _ 

,  _Li  '  —  I 

n=oo  ^ 

by  Theorem  c. 

Thus  it  is  seen  that  all*  of  these  definitions  have  properties 
(i)  to  (iii)  in  common.  We  can  accordingly  generalize  our  notion 
of  summability  by  stating  a  definition  in  terms  of  these  properties 
themselves. 

Definition;  A  series  shall  he  said  to  be  A-evalnable,\  and  to  have 
the  sum  s  whenever  the  following  conditions  are  fulfilled  : 


*  Except  definition  (10) 

t  We  shall  hereafter  use  the  term  evaluable  in  the  case  of  definitions  in 
terms  of  properties  of  general  functional  coefficients  of  the  st;  the  word  suni- 
mahle  we  shall  retain  for  concrete  definitions  with  specific  coefficients. 


(A) 


DEFINITION   OF   SUM    OF   A   DIVERGENT   SERIES 

(i)      Ij  ai(n)  =  o,  for  fixed  i, 

71=00 

(ii)      L  [ai{n)  +  ao{n)  +  •  •  •  +  a„(w)]  =  i, 

n=:oo 

(iii)     a,(w)  >  o, 

(iv)      Tj  [aiin)si  -\-  02(^)52  +  •  •  •  +  an{n)sn]  =  s. 


49 


We  shall   now  justify  this  definition  by  proving  the  following 
theorem : 

Theorem  ii  :  If  a  series  is  convergent  then  it  is  A-evalnahle* 

By  (iv)  we  may  write: 


(v) 


[ai{n)  +  Giin)  +  •  •  •  +  an{n)]  +  r„  =  i, 


Now,  by  (v), 

I  cii{n)si  +  ai{n)s2  +  •  •  •  +  an{n)sn  —  s  \ 

=  I  {ai(n)si  +  a2{n)s2  +  •  •  •  +  anin)Sn} 
-  (ai(«)  4-  a2(n)  +  •  •  •  +  a„(w)  +  r„)5  ] 

<  I  ai{n){si  -  s)  +  a2in)(s2  -  5)  +  •  •  •  +  ap{n){sp  -  s)  \ 
+  1  ap+i{n){sp+i  -  s)  +  •  •  •  +  a„(w)(5„  -  5)  1  +  |  r„5  |  . 

Since  the  series  is  convergent,  we  can  choose  i  so  large  that 

\si  -  5I  <  ?7,         i  >  p. 

Let  /  be  the  largest  of  the  numbers  \  Si  —  s\,  (or  i  =  i,  2,  •••  p. 
We  have,  then, 

I  aiin)si  +  a2{n)s2  +  • ;  •  +  an{n)sn  —  s  \ 

<  {ai{n)  \sy  -  s\-\-  ■••  -\-  ap{n)  \sp  -  s\\ 

*  Theorem  1 1  obtains  if  condition  (iii)  is  replaced  by  the  broader  condition: 
|ai(w)l  +|a2(»)l  +  •••  +|a„(M)|  <  K. 


50  UNIVERSITY   OF  MISSOURI    STUDIES 

+  {aj,+i{7i)\sp+i  -  5l  +  •  •  •  +  an{n)\s„  -  5|}  +  \rns\ 

<  {aiin)-\ \-ap(n)}l+{ap+i{n)-\ \-an{n)]r]-\-\rns\ 

<  81  +  v  +  \rns\,     n>  N* 


=  e. 


Hence 


Tj  [ai{n)si  +  a2{n)s2  +  •  •  •  +  a„(n)Sn]  =  s. 


Our  definition  of  i4-evaluability  is  now  justified. 

The  question  naturally  suggests  itself  as  to  whether  for  a 
sequence  (5„)  which  diverges  to  +  00, 

n 

Jj  ]C«.(w)5i  =  +  00. 

The  answer,  which  is  in  the  afhrmative,  is  embodied  in  the  fol- 
lowing theorem : 
Theorem  iia:  // 

Ij5„   =    +   00, 
n=oo 

and  conditions  (i),  (ii),  (iii)  are  satisfied,  then 

11 
Ij  ^ai{n)Si  =  +  00. 

n=QO  i=l 

By  hypothesis,  Sn  >  N,  n  >  m.     Hence 

n  m  n 

On  =  ^ai{n)Si  =  '^ai{n)si  +  2J  ai{n)si 

i=l  i=l  »i+l 

m  n 

>  ^ai{n)Si  +  //   X)    «»■(«)• 

*  By  (i),  [ai(«)  +  •  •  •  +  ap{n)]  <  8,  n  >  N,  p  having  been  chosen  first, 
and  then  held  fast.  By  (iii),  [ap+i{n)  +  •  •  •  +  c„(n)]  <  [ai(n)  +  •  •  •  +  a„(w)] 
<  I  by  (ii). 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES 


51 


Since 


[m.  n  -j 

T,ai(n)Si  +  N   Y.   a.(n)     =  N, 


it  follows  that 


Minimum  Ij  cr„  >  iV; 


and  since  N  is  an  arbitrary  number, 

We  have  seen  that  the  generalized  definition  includes  a  large 
number  of  the  specific  definitions  of  summability  which  we  have 
considered.  But  we  see  now  that  if  we  take  any  functions 
whatever  for  ai{?i),  subject  merely  to  the  restrictions  (i),  (ii) 
and  (iii),  we  may  obtain  a  possible  definition  of  summability. 
Thus,  we  may  take  as  our  definition,  for  example. 


(II)  5=L 


^1+7  + 
2 


+ 


log  n 


Sl-\--S2-\- 
2 


-\--Sn 

n 


!+,+ 


+ 


71       -^ 


This  formula  is  of  interest  to  us,  since  it  affords  an  example  of 
a  definition  which  is  broader  than  Ces^ro-summability  of  order  i, 
and  yet  perhaps  not  so  general  as  that  of  order  2.  For  since 
i/w  steadily  decreases,  it  follows  from  Theorem  8  that  formula 
(11)  gives  a  value  to  all  series  that  are  Ces^ro-summable  of 
order  i,  and  that  these  values  are  the  same  for  both  definitions. 
That  (11)  is  really  more  general  than  summability  of  order  i 
follows  from  the  example  i— 3  +  5  —  7  +  ----  This  series 
is  not  summable  of  order  i ,  since 


—  +  o; 

■n-—oa   ^ 

however  we  obtain  from  (11),  for  the  corresponding  sequence, 


52 


UNIVERSITY   OF   MISSOURI    STUDIES 


[' 


-   I   +   I    -   I 


J  n=«  L   log  W   J 


log  n 


Nevertheless,  (ii)  is  probably  not  equivalent  to  summability 
of  order  2,  as  the  following  reasoning  suggests.  A  necessary 
condition  that  a  series  give  a  result  by  (11)  is 


Un 


n=oo  n  log  n 


=  o. 


This  is  not,  however,  a  necessary  condition  for  summability  of 
order  2  f — so  that  we  might  find  a  series  for  which 


«=«  n  log  n 


+  0, 


which  is  nevertheless  summable  of  order  2. 

We  have  seen  that  the  .4 -definition  includes  most  of  the  cases 
of  summability  which  we  have  discussed,  but  we  have  been 
obliged  to  omit  Borel's  definitions.  In  order  to  include  the  Borel- 
mean-definition,  we  shall  now  generalize  Theorem  1 1 ,  as  well  as  the 
definition  which  we  have  based  upon  it.  Replacing  a,(«) 
by  ai(a),  where  a  may  be  independent  of  Ji,  Theorem  (11)  may 
be  stated  in  a  more  general  form: 

Theorem  12:  From  the  conditions: 


*o=  L. 

n=oo 


S\  +  5^2+   •"    -\ 7-  .Jn  +  l        5i  +^52+    •  •  •    +  '-Sn 

n  -\-\  n 


i+h  +  •••  + 


n  +1 


u„ 


1+1  + 

Sn   —  Sn-l 


+  - 


Sn 


n  =  oon  log, 


„=«,«log„        n=oo      nlogn 

t  A  necessary'  condition  for  summability  of  order  2  is 

tin 


See  p.  10. 


7i  =  00   W 


DEFINITION   OF   SUM   OF   A  DIVERGENT   SERIES 

(i)     L  ai(a)  —  o  for  fixed  i, 
0=00 

(ii)     L  [ai{a)  +  a2{a)  +  •  •  •  +  a„(a)]  =  i, 

n=ao 

(iii)     ai(a)  >  o, 

(iv)       1j  Sn   =  S, 


53 


may  be  deduced  the  result: 

Tj   Ij  [ai{a)si  4-  a2ia)s2  + 


a=oo  n=(» 


+  a„(Q!)5„]  =  5. 


We  shall  first  show  that 

Tj  [ai(a)si  +  a2(a)s2  +  •  •  •  +  ar,{a)Sn] 

TO=00 

exists  for  every  definite  a.     Taking  a  definite  value  of  a, 
\a„{a)Sn  +  an+i{a)Sn+i  +  •  •  •  +  a„+p(«)5„+p| 

<   an{a)\Sn\   +    •  •  •    +  an+p{a)\Sn+r 


<j'Ahy  (ii) 


{(n  >  N,  anyp)) 


=  e. 


Hence 


Zlanioc)Sn 
n=l 

converges  for  every  value  of  a.     Since 

zlan{cx)Sn 

has  a  sense,  we  may  write: 


z2an{oi)Sn  —  S 


X  an{oi)Sn  —  2  «n(a)  '  s\  by  (ii) 


54 


UNIVERSITY   OF   MISSOURI    STUDIES 


53  On  (a)  (^n  —  s) 


m—l 


< 


^an{a){s„  —  s) 


+ 


S  an{a){Sn-s) 


<  H^an(a)  +  e, 


since   \sn  -  s\  <  e,  n  >  m,  and   \sn  -  s\  <  H,  n  <  m  by   (iv). 
Since,  however, 

TO— 1 


by  (i),  it  follows  that: 


Maximum  Ij 


^an{a)Sn—S 


<e  -\-  Maximum  J^  Hz^  CLnict)  =^- 

0=00        n=l 


Since  e  is  arbitrarily  small,  the  maximum  limit  on  the  left  must 
be  zero,  and  therefore  the  actual  limit  is  zero,  i.  e., 

00 

Ij  53  an{a)Sn  =  s. 

a=oo  n=l 

It  is  readily  seen  that  Borel's  mean-definition  satisfies  con- 
ditions (i)  to  (iii)  of  Theorem  I2.  For  we  have,  in  satisfaction 
of  condition  (i), 

L^=o; 

0=00    '-■ 

that  condition  (ii)  is  satisfied  follows  since 


a       a'' 

^+^  +  2-.+ 


+ 


=  i; 

and  finally,  since  a^/e"  >  o  for  a  >  o,  it  follows  that   (iii)  is 
fulfilled. 

We  might  accordingly  generalize  our  definition  of  evaluability, 
to  include  Borel's  mean-definition,  by  using  the  hypotheses  (i) 
to  (iii)  of  Theorem  I2  as  a  basis.  It  turns  out,  however,  that 
we  may  generalize  Theorem  I2  still  further,  and  that  we  can 
accordingly  obtain  a  still  more  general  definition  of  evaluability. 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES  55 

Let  US  take  as  coefficients  of  the  5,  functions  of  both  n  and  a, 
and  write: 

(i)     J^  ai{a,  n)  =  o, 

n=oo 

n 

(ii)     L  S  cii{a,  n)  =  I, 

n=oo  t=0 

(iii)     ai(a,  n)  >  o. 

If  now  these  conditions  are  fulfilled  for  a  fixed  value  of  a, 
and  if 

JLJ  Sn    =    S, 
n=oo 

it  follows  from  Theorem  11,  that 

n 

Ij    2  Oi(«,  W)5t   =  5. 
n=oo  i=0 

Since  this  limit  exists  for  every  value  of  a,  under  our  hypothesis, 
we  may  write: 

n 

(iv)  L   Li  S  Oi(a,  n)si  =  s, 

a=oo  n—co  i=0 

and  a  definition  that  readily  suggests  itself,  even  when  the  series 
is  not  convergent,  is  that  conditions  (i)  to  (iv)  be  fulfilled. 

We  have  demanded  at  the  very  start,  however,  that  every 
definition  should  satisfy  certain  fundamental  requirements, 
which  we  have  enumerated  on  page  2,  and  while  the  definition 
proposed  does  fulfil  the  first  two  of  those  requirements,  as  we 
have  just  seen,  it  does  not  fulfil  the  third  requirement*  without 
further  restrictions  on  the  coefficients.! 

Our  third  fundamental  demand  was  that  when  the  series 
«o  +  Wi  +  W2  +  •  •  •  +  w„  +  •  •  •  has  the  value  s,  then  the 
series  Wi  +  W2  +  •  •  •  +  «n  +  •  •  •  must  have  the  value  s  —  Uo; 

*  The  same  is  true,  of  course,  for  the  ^-definition;  we  have  deferred  the 
similar  considerations  for  that  case,  since  they  may  be  included  under  this 
more  general  one. 

t  It  is  obvious  that  the  fourth  and  fifth  requirements  are  also  fulfilled. 


56  UNIVERSITY   OF   MISSOURI   STUDIES 

or  stated  in  terms  of  sequences,  if  Sn  =  Uq  -\-  Ui  -\-  •  •  •  +  w„, 
when  the  sequence  So,  Si,  52,  •  •  •  Sn,  •  •  •  has  the  value  s,  then  the 
sequence  Si  —  Uo,  S2  —  Uo,  •  -  •  Sn  —  Uo,  •  •  -  has  the  value  s  —  Wo- 
lf we  assume,  for  the  moment,  that  whenever  either  one  of  the 
two  sequences 

•So»    -^1.    S2,     •  •  '   Sn,     •  '  ■ 

^1)    ^2,     '  '  '   Sn,      '  '  ' 

has  the  value  s,  the  other  does  also;  then  we  shall  satisfy  our 
third  requirement  if  we  prove  that  whenever  Si,  S2,  ss,  •  •  •  Sn,  •  •  • 
has  the  value  s,  then  ^1  —  Uo,  S2  —  Uo,  Sz  —  Uo,  •  •  •  5„  —  Mq,  •  •  • 
has  the  value  5  —  Wo-  Now  this  it  is  easy  to  prove.  For  we 
have  by  iv,  p.  55, 

n  n 

Jj  Ij  Y1  ai{c(,n){si-th)=  J^   Tj^  aiia,n)Si-Uo  =  s  —  Uo 

0=00  n=m  i—0  a=oo  n=oo  i=0 

by  (ii),  p.  55. 

It  remains  then  to  consider  under  what  restrictions  we  can  justify 
our  assumption  that  the  two  sequences 

So,    Si,    52,     •  •  •    Sn,     '  '  ' 
S\,    S2,     '  •  •   Sfi,     '  '  ' 

always  have  a  value  together.     To  get  an  idea  as  to  the  nature 
of  the  condition  which  we  shall  have  to  add,  let  us  consider,  for 
concreteness,  what  happens  in  the  case  of  Borel's  mean-definition. 
Using  the  notation  of  page  12,  we  have: 

a  a-  a" 

S{a)    =   So  +  Si-  -\-  S2—.-{-    '■'    +  Sn—.+       ", 

I  2 !  n  I 


a 


.n-l 


S\a)    =    5l   +  52  -  4-    •  •  ■    +  Sn  7- "Vj  + 

I  [n  —  i)\ 


a  d"  oi"'  ^ 

s'{a)  -  s{a)  =  Ui  +  «2  -  +  «3  --,  +  •  •  •   +  Un  7 -T,  + 

I  2  !  \n  —  I^  ! 

Borel's  definition  being 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES  57 

If  we  assume*  that  L  s{a)  =  oo, 

a— CO 

we  have  an  indeterminate  form,  so  that 


L  s'(a)  —  s(a) 


or 


=  o, 
which  may  be  written, 

T        T  a  r  .  <^  a^  0!"~| 

It  is  accordingly  suggested  that  we  assume,  in  general, 

(v)      L   Ij  [ao(a,  n)ui  +  01(0;,  «)w2  +  •  •  •  +  fln(a,  w)w„+i]  =  O. 

o=ao  n=oo 

As  a  matter  of  fact,  this  condition  is  sufficient,!  for,  from  (iv) 
(iv)      Ij   Ij  [aoia,  n)so  +  ai{a,  n)si  +  •  •  •  +  an(a,  n)sn]  =  s  and 

a=ao  n=oo 

adding  (iv)  and  (v)  we  obtain 

Ij    Ij[ao(Q:,  n)si  +  ai{a,  ?i)s2  +  •  •  ■  +  an{a,  n)sn+i]  =  s, 

a=ao  n=oo 

which  proves  that  when  the  sequence  5o,  Si,  •  •  •  5„,  •  •  •  is  eval- 
uable  to  s,  so  is  the  sequence  ^i,  S2,  '  -  •  Sn,  •  -  -  •  By  subtracting 
(v)  from  the  last  limit  we  show  in  the  same  way  that  when  the 
sequence  Si,  Sz,  •  •  •  5„,  •  •  •  is  evaluable  to  s,  so  is  the  sequence 
^0,  ^i»  ^2,  '  •  •  Sn,  •  •  • .  Thus,  condition  (v)  causes  our  definition 
to  satisfy  the  third  requirement  of  page  2.  If  we  wish  to  be 
able  to  drop  any  finite  number  of  terms,  we  shall  have  to 
require  a  condition  more  general  than  (v),  as  we  shall  do  in  the 
following  definition: 


*  This  assumption  is  not  essential,  since  our  object  is  simply  to  arrive  at  a 
certain  condition  on  the  a,(a,  n). 

t  Condition  (v)  is  not  satisfactory  since  it  is  a  condition  on  the  sequence, 
as  well  as  on  ai{n,  a).  It  would  be  desirable  to  have  on  ai{n,  a)  further  re- 
strictions, sufficient  to  cause  (v)  to  hold  for  all  sequences. 


58 


UNIVERSITY   OF  MISSOURI    STUDIES 


Definition:  A  series  shall  he  said  to  he  B-evaluahle  and  to  have 
the  sum  s  whenever  the  following  conditions  are  fulfilled: 

(i)  Ij  a.(a,  n)  =  o, 

n 

(ii)  L  ^ai{a,  w)  ^  I, 


B 


n=oo  1=0 

(iii)  ai{a,  n)  >  o, 

n 

(iv)  L   Ij  Yioiia,  7t)si  =  s, 

a=oo  n=oo  i=0 
n 

(v)  Ij    L  S«i(Q:,  w)m,+A;  =  0,      ^   =    I,  2,   •  •  •  /). 

a=oo  n=ao  »=0 


We  have  seen  that  this  definition  includes  all  of  the  definitions 
of  summability  which  we  have  considered,  except  possibly  the 
Borel-integral  definition.  We  have  not  yet  subjected  this 
integral  definition  to  the  test  of  our  fundamental  requirements; 
let  us  now  do  this. 

That  requirements  (i)  and  (ii)  are  satisfied  follows  from  the 
following  theorem:*  If 

Ij  Sn  =  s, 


then 


where 


/»C0 

Jo 


e~''u{r)dr  =  s, 


u{r)  =  Ua  -\-  Ui--\-U2—  + 


+  M„  — ,  + 


It  is  obvious,  too,  that  requirements  (iv)  and  (v)  are  satisfied.  Let 
us  accordingly  limit  our  considerations  to  requirement  (iii) .  With 
regard  to  this  requirement  we  have  the  following  state  of  affairs  rf 

*  Hardy:  Quarterly  Journal,  Vol.  35,  p.  22;  Bromwich,  loc.  cit.,  p.  269. 

t  The  quotation  is  taken  from  Bromwich,  loc.  cit.,  p.  271.  The  first  of  the 
propositions  was  proved  by  Borel,  loc.  cit.,  p.  lOi;  Hardy  proved  the  second 
proposition  by  an  example:  Quarterly  Journal,  Vol.  35  (1903),  p.  30. 


DEFINITION  OF   SUM   OF  A  DIVERGENT   SERIES  59 

"Any  finite  number  of  terms  may  be  prefixed  to  a  summable 
series,  and  the  series  will  remain  summable.  .  .  .  But  the  removal 
of  even  a  single  term  from  the  beginning  of  the  series  may  destroy 
the  property  of  summability." 

Inasmuch  then  as  the  integral-definition  fails  to  satisfy  one 
of  our  fundamental  requirements,  we  are  obliged  to  rule  it  out. 
In  fact  Borel  himself  ruled  it  out,*  replacing  it  by  absolute 
summahility .'\  This  definition  does  satisfy  requirement  (iii), 
as  Borel  proves,  J  and  it  obviously  satisfies  requirements  (ii), 
(iv)  and  (v).  Furthermore,  Borel  makes  the  statementi  that 
convergent  series  are  always  absolutely  summable.  Hence  it 
would  follow  that  the  definition  of  absolute  summability  is  to 
be  retained,  since  it  seems  to  satisfy  all  of  the  fundamental 
requirements. 

But  Borel's  statement  that  convergent  series  are  always 
absolutely  summable,  is  incorrect,  as  Hardy  §  has  shown  by  the 
following  example: 

Un  = -. — ,     n  =1^, 

y-Un  =  o,  n  not  a  square. 

In  fact  the  series  in  question : 

-1+0  +  0  +  1  +  0  +  0  +  0  +  0-^+  ••• 

is  convergent,  while 

e~'^\u{r)\dr 


f 

t/o 


is  divergent.     Thus,  since  absolute  summability  fails  to  satisfy 

*Loc.  cit.,  p.  99. 
t  See  p.  14. 
X  Loc.  cit.,  p.  100. 
§  Hardy,  loc.  cit. 


60  UNIVERSITY   OF   MISSOURI    STUDIES 

the  first  fundamental  requirement,  this  definition  too  cannot  be 
retained.* 

We  have  seen  that  the  J5-definition  satisfies  all  of  our  funda- 
mental requirements,  and  that  it  includes  as  special  cases  all 
of  the  proposed  definitions  of  summability  which  satisfy  those 
requirements.  Our  definition  of  5-summabiIity  is  accordingly 
justified. 

We  proceed  to  the  statement  of  the  following  definitions: 
Definition  i :  A  series  shall  he  called  abstractly-evaluable,  and 
to  have  the  value  s,  if  the  following  conditions  are  fulfilled : 

(a)  L  [ai{n)si  +  a2(n)s2  +  •  •  •  +  a„(n)sn\  =  s, 

n=oo 

(b)     the  fundamental  requirements  of  page  2  are  satisfied. 
Definition   2:  An  abstractly-evaluable  series  of  functions  of  a 
variable  shall  be  called  uniformly  evaluable,  if: 

L  [fli(w)5i(x)  +  a2{n)s2{x)  +  •  •  •  +  an{n)sn{x)] 

=  L  f{x,  n)  =  s{x) 

n—x> 

tmiformly. 

From  these  definitions  follow  at  once  several  theorems. 

Theorem  13:  A  uniformly  evaluable  series  of  continuous 
functions  represents  a  continuous  function. '\ 

For  f{x,  n)  —  ai{n)si(x)  +  •  •  •  +  an{n)snix)  is  a  continuous 
function  of  x;  and  since 

Hjfix,  n)  =  s{x) 

'H  —  ao 

uniformly,  it  folloAVS  that  s{x)  is  continuous. 

Similarly,  we  should  obtain  in  the  usual  way,  the  following 
two  propositions: 

*  It  is  for  this  reason  that  we  omit  from  further  considerations  the  integral 
definition  and  the  extended  definitions  given  by  Borel  himself  and  by  Le  Roy. 
See  p.  14,  supra. 

t  The  same  proof  applies  when  the  continuity  is  with  respect  to  some 
assemblage. 


DEFINITION  OF   SUM   OF   A   DIVERGENT   SERIES  6 1 

Theorem  13A:  ^  sufficient  condition  that  an  abstractly-evaluable 
series  of  continuous  functions  represent  a  continuous  function  is 
that  the  related  sequence,  f(x,  n),  have  Dini's  simple-uniform  con- 
vergence* 

Theorem  13B:  A  necessary  and  sufficient  condition  that  an  ab- 
stractly-evaluable series  of  continuous  functions  define  a  continuous 
function  is  that  fix,  n)  have  ArzelcL's  quasi-uniform  convergence.] 

Theorem  14:  A  uniformly  evaluable  series  of  continuous 
functions  may  be  integrated  term  by  term. 

We  wish  to  prove  in  this  case  that 

I    Li  [ai(w)5i(x)  +  a2{n)s2{x)  +  •  •  •  +  an(n)sn(x)]dx 

da    n=ao 

=  Ij    I    [ai(n)si{x)  +  ao{n)s2(x)  +  •  •  •  +  an{n)sn{x)]dx 
or 

nb  pb 

I    1jf{x,  n)dx  =  Ij    I   /(x,  n)dx, 

but  this  equation  is  precisely  a  statement  of  the  theorem  that 
a  uniformly  convergent  sequence  of  continuous  functions  may  be 
integrated  term  by  term. 

Theorem  15:  If  a  series  of  continuous  functions  is  convergent 
for  all  values  of  x  in  an  interval,  except  possibly  for  x  =  x^;  and 
if  two  sets  of  functions  ai(n),  bi{n)  render  the  series  abstractly- 
evaluable  at  Xo,  to  the  values  s  and  t  respectively;  then,  if  the  evalua- 
bility  of  each  of  the  definitions  is  uniform  in  the  interval,  then  s  =  t. 

Letting 

n 

f{x,  n)  =  '^a^{n)Si{x), 
and 

n 

g{x,  n)  =  ^bi(it)si(x), 


*  Dini:  Fundamenli  per  la  teoretica  delle  Funzioni  di  variabili  reali.     Pise, 
1878,  p.  103. 

t  Arzelcl:  Mcmoires  de  Bologne,  1899. 


62  UNIVERSITY   OF   MISSOURI    STUDIES 

and  remembering  that  since  the  series  is  convergent,  x  +  :^o, 

it  is  true  that 

Li/(.T,  n)  =  Ijg(^,  n),     X  4=  Xo, 

n=oo  re:=oo 

we  have  from  the  uniformity, 

Li  Jjf{x,  n)  =  Jjfixo,  n)  =  sA 

L/  L  g(^,  «)  =  L  §(^0,  n)  =,i   I 

and  hence  s  =  t. 

We  may  obviously  state  the  preceding  theorem  in  the  following 
more  general  manner: 

Theorem  15A:  //  a  series  of  junctions  continuous  on  an  as- 
semblage (E)  is  cofivergent  at  all  points  of  (E),  except  possibly  at 
X  =  .To,  which  is  a  limit  point  of  (E);  and  if  two  sets  of  functions 
diin),  hi{n)  render  the  series  abstractly-evaluable  at  xo,  to  the  values 
s  and  t  respectively;  the?i,  if  the  evaluability  of  each  of  the  definitions 
is  uniform  on  {E) ,  it  follows  that  s  =  t. 


§  7-   APPLICATIONS 

We  shall  first  consider  an  application  of  the  definition  of 
abstract  evaluability  to  integral  equations,  and  we  shall  obtain  a 
generalization*  of  a  theorem  due  to  Vol  terra,  f  Let  us  seek  for 
a  continuous  solution  of  the  integral  equation, 

u{x)  =Kx)+  f  K{x,  ^)uWd^, 

where  K{x,  y)  is  continuous,! 

{a<x<b] 
I  a  <  3'  <  6  J 

and  f(x)  is  continuous,  a  <,  x  <_  b. 

Following  the  method  of  Volterra,  we  shall  form  the  iterated 
functions: 

rKi{x,  y)  =  K{x,  y), 

(12)  \  r^ 

[K,{x,  y)  =J    K,{x,  OKi-^U,y)d^. 

Then 

Ki(x,  y)  =   \    K{x,  ^{)K{^u  h)  -  -  ■  K{U-i,  y)dl^i.,  -  •  •  d^. 


and 


Ki+iix,  y)=  f  Ki(x,  k)KM.  y)dk- 


*  Our  result  is  more  general  if  we  restrict  ourselves  to  Volterra's  method; 
a  much  more  general  result  has  been  obtained  by  Fredholm  by  means  of  a 
different  method.     See  Acta  Math.,  Vol.  27  (1903),  p.  365. 

t  Rendiconti,  Accademie  del  Lined,  series  5,  Vol.  5,  1896. 

X  The  theorem  can  be  proved  with  much  broader  restrictions  on  K{,x,  y). 

63 


64  UNIVERSITY   OF   MISSOURI   STUDIES 

If  we  first  put  i  =  I,  i  -{- j  =  m  in  this  formula,  and  then  put 
j  =  I,  i  -\-  j  =  m,  we  obtain:* 

(13a) 


(13&) 


Km{x,  y)  =    f  Km-iix,  OK^a,  y)dl 


Volterra  now  proves  that  if  the  series  Ki{x,  y)  +  •  •  •  +  Kn{x,  y) 
+  •  •  •  converges  uniformly  in  s,  then  the  integral  equation  has 
one  and  only  one  continuous  solution.  We  shall  prove,  more 
generally,  the  following  theorem: 

Theorem  16:  7/  the  series  Ki{x,  y)  +  •  •  •  -{■  Knix,  y)  -\-  -  •  •  is 
uniformly  evaluable  in  the  abstract  sense,  then  the  integral  equation 
has  one  and  only  one  continuous  solution. 

Since  21  Ki{x,  y)  is  evaluable, 

-  ki^,  y)  =  Xi(^,  y)  +  K,{^,  y)+  ■■■  +  K^U,  y)  +  ---, 
and  by  our  fundamental  requirement  (v),  p.  2, 

-  k{^,  y)K,{x,  0  =  K,ix,  OKii^,  y)  +  ^iC-^.  ?)^2(^,  3')  +  •  •  • 

+  i^i(x,  OKn(^,y)  +  .... 

Moreover,  the  last  series  is  uniformly^  evaluable. 

Hence  we  may  integrate  term  by  term,  by  Theorem  14,  obtain- 
ing 

-  r  K{x,  ^)k{^,  y)d^  =   f  K,{x,  ^)Xi(?,  y)d^ 

Jr>b  r*b 

'    K,{x,  ^)K2a,  y)d^  +  . .  •  +  I    K,{x,  k)Kn{^,  y¥k  +  •  •  • 

=  K,{x,  y)  +  Kz{x,  >')+•••+  Kn+i{x,  y) -\-  -  ■  ■ 


*  The  first  of  these  two  formulae  is  the  same  as  the  definition  of  Kmix,  y). 
t  The  uniform  evaluability  can  be  established  in  precisely  the   same  way 
as  in  the  case  of  convergence. 


DEFINITION   OF   SUM   OF   A  DIVERGENT   SERIES  65 


by    (i3«)-      The     series    last    considered    has    for    its   value, 
—  k{x,  y)  —  K\{x,  y)  so  that 

K{x,  ^)k{^,  y)d^  =  K(x,  y)  +  k{x,  y). 


f 


By  using  (13&)  in  a  similar  fashion, 

W.  ^)K{^,  y)d^  =  K{x,  y)  +  k{x,  y). 


f 


The  rest  of  the  proof  is  the  same  as  that  given  by  Volterra,* 
who  obtains  as  the  unique  continuous  solution: 

(14)  u{x)  =Kx)-    f  Hx,  mOd^. 

It  is  not  difificult  to  construct  an  example  for  which  the  series 
K.i{x,  y)  -\-  '  '  '  +  Kn{x,  y)  -\-  •  •  •  does  not  converge  but  is, 
for  example,  Ces^ro-summable  of  order  i.  Let  us  look  for  a 
continuous  solution  of  the  integral  equation: 

2    n 
u{x)  =  I  +  ~  I     sin  (:k  —  y)n{y)dy. 

TT  Jo 

Here 

2  2 

Ki{x,  y)  =  -sin  (x  -  y),     K^ix,  y)  =  -  -cos  {x  -  y), 

TV  T 

2  2 

Ksix,  y)  =  -  -sin  (x  -  y),     K^ix,  y)  =  -cos  (x  -  y), 

and  so  on,  so  that  the  series  becomes 

-  k{x,  y)  =  T,Ki{x,  y) 

=     -  sin  (x  —  3')  —  -  cos  (x  —  y)     ( i  —  i  +  i  —  i  H ) , 

LtT  TT  J 

which  is  not  convergent.     Its  summable  value  (Ci)  is,  however, 
—  k{x,  y)  =  -[sin  {x  —  y)  —  cos  (x  —  y)] 

TT 


*  Volterra,  loc.  cit. 


66  UNIVERSITY   OF   MISSOURI    STUDIES 

SO  that  our  solution  will  be: 

I  r 

u(x)  =  I  +  -  I      [sin  (x  —  3')  —  cos  (x  —  y)]dy. 

TT  Jo 

An  interesting  application  of  Cesaro-summability  of  order  i 
has  been  given  by  L.  Fejer.*  It  is  well-known  that  if  a  function 
/(x)  satisfies  Dirichlet's  conditions,  it  may  be  developed  into  a 
convergent  Fourier  series.  Fejer  has  shown  that  if  f(x)  is  finite 
and  integrablef  and  of  period  2t,  then  the  Fourier  development 
corresponding  to  f{x)  will  be  Cesaro-summable  of  order  i  to  the 

value 

H/(^  +  o)+/(x-o)] 

at  all  points  at  which  the  function  is  continuous  or  has  a  finite 
jump.  A  similar  result  has  been  obtained  for  the  development 
in  terms  of  Bessel  functions  by  C.  N.  Moore. + 

We  proceed  to  the  consideration  of  a  similar  theorem  in  the 
case  of  the  development  of  a  function  in  terms  of  power  series. 
If  we  write: 


(15) 


^2  l,n-l 

Sn  =  m  +  hf{a)  +  -r'{a)  +  •  •  •  +  J^^^^r^J-Ka), 


Rn  =  f(a  +  h)  -  Sn, 
then  Taylor's  Series  with  a  remainder  may  be  written 

f{a   +  h)    =    Sn-\-  Rn, 

where  it  is  found,  on  the  assumption  thatf'(x),  •  •  •  f'-"'^{x)  exist, 
in  the  interval  (a,  a  -\-  h),  that  § 

(i6)  Rn  =  -J"{a  +  eh),     o  <  0  <  I. 

n ! 


*  Math.  Annalen,  Bd.  58,  1904,  p.  51. 
^  fix)  may  become  infinite  at  a  finite  number  of  points. 
t  Transactions,  Am.  Math.  Soc,  Vol.  10  (1909),  p.  391. 
§  This  is  Lagrange's  form  for  the  remainder.     See  Goursat-Hedrick,  loc. 
cit.,  p.  90. 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES 


67 


From  (15)  it  is  obvious  that 
(17)    f(a  +  h) 
if  and  only  if 


(17)    f{a  +  h)  =f{a)  +  hj'{a)  +-,/"(a)  + 


^2 
2 


+  Z1 /"(«)  + 


n=<x> 

If  it  should  turn  out  that 

IjRn  =  k  ^  o, 

n=ao 

then  it  follows  that  the  series  of  the  right  member  of  (17)  cannot 
represent  f{a-\-h).     But  if  Tj  Rn  does  not  exist,  though   the 

series  cannot  then  be  convergent,  it  may  be  possible  to  choose  a 
definition  of  sum  which  will  give  for  its  value  /(a  +  h).  Thus 
we  obtain  from  (15)  and  (16) 

-  E  5i  =  /(a  +  /j)  -  -  E  ^i  =  /(a  +  h)  -  R„, 


(18) 


Ri 


/(»)(a+  di)h, 


Rn    =  ~   Z^  Ri- 


As  before,  we  consider  three  possibilities.     If 

JjRn  =  o, 

n=ao 

then 

I    " 

1j  -'^  Si  =  f{a  -\-  h); 

n=co  '^  t=l 
if 

I      ** 

J^Rn  =  k  ^-o,     L  -  Z  ^i  +  /(a  +  h); 

I      " 

and  if  L  i?„  does  not  exist,    L  -  XI  -Jt  does  not  exist. 

This  result  is  not  satisfactory  as  it  stands,  however,  because 
of  the  di  which  appear  in  (18),  and  which  may  dififer  with  i. 


68  UNIVERSITY  OF   MISSOURI    STUDIES 

We  shall  accordingly  proceed  to  obtain  another  form  for  R„. 
We  have: 

nf(a)  +  («  -  i)/'(a)  ~  +  (n  -2)f"{a)  ^  +  •  •  • 
(iq)                                       .                    h"-^                       ;z"-i 
-  2^  Si  = . 

w  <=i  n 

For  fixed  a  and  h  we  let  the  difference 

I  -A*  h^ 

n  1=1  P 

and  we  consider  the  auxiliary  function 

^{x)  =  1 1  nf{a  +  /O  -  [^/(.t)  +  (n  -  i)  ^^^^=-^^/'(.t) 

+  (n-2)  ^    ^^,       V"W  +  ■  ■  ■  +  2      ^,_,)!      /-'W 

,  (a  +  /'-.x-) '"-",,„.„,  .    ,   (a  +  h-x)' 
+  "     („-,)!       /'"  "W  + ^ "P 

Since    <p(a)  =  ^(a  +  h)  =  o,    it    follows    that    <p'{a  +  6//)  =  o, 
o  <  0  <  I.     But 

n<p'{x)  =  -  ^nfix)  +  {n-i)  ^^-^^^f^fix) 

+  («-2) -^ /    (x)  +  ---+       (^_i),     /H^)J 

+  [(w  -  i)/'W  +  (tz  -  2)  (^+^^^^-^)^.(^) 

+  (w  -  3)  ^^  +  ^^~^)y/(-,.)  +  . . .  +  (^  + ;,  _.-,)P-i„p 

(a  +  //  —  .t)"-i  1 

+     ^,-1)1     /"C^)  -  («  +  /^  -  -t)-'«P J  . 


] 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES  69 

Since  <p'{a-\-6h)  =  o, 

where 

^  =  a  +  0/;,     a  +  /?  -  ^  =  /z(i  -  0)  =  X,     o  <  0  <  i. 
If  we  choose  />  =  i,  we  obtain: 


(20) 


lW)-^\f"i^)^l. 


Rn  =  hP  =  -\  no  +  7/"  (^)  +  r,/'"(^)  + 


+  zr^-TTiZ-K^)  +  7rz-7y!/"(^) } 


± fn-Ut)    +  -A 


If  now 


then 


-  Z  ^f  =  /(a  +  h)  -  i?„, 


7i=a>  f^  4=1 


if  and  only  if 


I    ** 

Ij  i?n    =   O. 


We  have  thus  proved : 

Theorem  17:  //  the  first  n  derivatives  of  f(x)   exist  in  the 
interval  {a,  a  -\-  h),  then 

/(a  +  h)  =  f(a)  +  -  /'(a)  +  ^,  f"(a)  +  .  . .  +  ^"  /-(a)  +  •  •  • , 
where  the  infinite  series  is  Cesdro-summable  of  order  i,  provided 

luRn    =   O, 

n=ao 

where 


70  UNIVERSITY   OF  MISSOURI    STUDIES 


«.=:.]  /'({)  +  7/"(»  +-yf"'i&  + 


^  =  a  +  eh,     \  =  a  -{■  h  -  ^,     o  <  6  <  i. 
Turning  now  to  the  v^-definition,* 


zL  (PiSi 


95 


—    Li        n 


we  may  obtain  a  form  for  the  remainder  similar  to  (20).     We 
shall  put 

n 

2-1  ^i    —    fin 

j=i 

and  obtain 

n 

.S^'^'      I  r  /f2 

~      K0)ipin   +  hf\a)ip2n  +  —J"{a)nn  + 
In    L  '' 


fin   L  2 

r   fi 

i=l 


(w  -  I)! 
We  now  define  P  by  the  relation: 


+  zr— TV,/"~Ka)^n„J 


fia+h)  -— =  -P  =  Rn 

fin  P 


*  This  definition  is  the  same  as  that  on  p.  37,  since 

n 

because 

l*    ^n  =  I. 

n=oo 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  7 1 

and  we  construct  the  function: 

<P(X)    =  \  <plnf(a  +  h)   -\   <pi„f(x)    +   (P2n  —, /'(X) 

(a-{- h  —  xY  ,,,,  ,  (a -{- h  —  x)""'^ 

+  ^3n  ' ^r-^f"(x)+  •  •  •   +  ^„„  (n-l)\  -^""'^^^ 


(a  +  h- 


^y       pi  I 


Since   <p(a)  =  (p{a-^  h)=o,  we   must   have     <p\^)=o,    ^  =  a-{-6h, 
o  <  ^  <  I.     But 


«pi 


nip  {x)=-{  iPlf'ix)  +  ip2 f'\x)  +  953  —^ f      (X) 


H +  <Pn  '    '(n-i)\     -^"^^^  -  (a  +  /^  -  A:)p-Vini'  [ 

so  that: 


(w-i)!     ' 

•pin    I  I  !  2 ! 

and  accordingly,  if  />  =  i, 

(21)  i?„  =  -:J^-  [  <p,fa)  +  ^2^r(^)  +  m-^ra)  +  •  •  • 

We  now  turn  our  attention  to  the  form  of  i?„  for  the  A- 
definition.     We  set 

n 

^  aj{n)  =  ain 
and  obtain: 

X)  ai(n)si  =  }{a)ain  +  hj\a)a2n  +  •  •  •  +  } ;T7/''~H^)'^n»- 

t=i  [n  —  \)\ 


72  UNIVERSITY   OF   MISSOURI    STUDIES 

We  define  P  by  the  relation 

«in/(a  +  h)  -  ^ainSi  =  hP  =  Rn* 
and  we  form: 

fi                           d  ~\~  h  —  X 
ainfia  +  /O  -     ainfix)  +  a^n —^ f'(x) 

1         (a-\-h-x)\,,^  (a±h-x)--' 

+  asn -j f"(x)    +    •  •  •    +  Qnn } ;TT— /""H^) 

2 !  {n —  I  j  I 

-\-(a-^h-x)P^  \. 

Since  (p(a)  =  (p{a  +  h)  =  o,  we  have  for  x  =  a  +  6h  —  ^, 


o  =  <p'(x)  =  -  I  a,{n)r{x)  -\-a,(n)"^^^^y-^f"(x)  + 


(w  -  i)! 


+  an(ii)^—iz 7TT— /"W  -^J' 


so  that  if,  as  before,  h(i  —  0)  =  X, 

X«-i 


P  =  ai(n)f'{0  +  a2(«)  r./'C^)  +  ^M  -J"'{^)  + 


+  «»(^irT)y"(^)' 


and  accordingly, 


i?„  =  /^[ai(w)/'(^)  +  a,{ii)y^S"{^)  +  a3(«)^/'"(^)  + 


+  an{n),J'     .,,/^"K^)]- 


(w  -  i)!- 


We  may  now  state  our  result  so  as  to  include  Theorem  17  as  a 
special  case. 

Theorem  18:  IJ  the  first  n  derivatives  of  f{x)  exist  in  the  interval 
{a,  a  -\-  h),  then 

*  We  previously  assumed   the  form  {hPJp)P  and  found   />  =  l  most  con- 
venient; we  here  choose  p  =  I  at  the  outset. 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  73 

f(a  -i-h)  =  f(a)  +  ~^f'(a)  +  ^/"(a)  +  •  •  •  +  -,Ma)  +  •  •  •, 

the  infinite  series  being  A-evahiable,  provided 

J^Rn  =  o, 

n=oo 

where 

R,  =  h^a,{n)n^)+a,{n)^f{^)  +  '  •  •  ^  ^M  ^^^^^H^)^ 

^  =  a+  eh,        \  =  a  -i-  h  -  ^,        o  <  0  <  i. 

We  proceed  now  to  the  proof  of  a  theorem  which  will  again 
illustrate  the  possibility  of  obtaining  results  from  very  general 
definitions. 

Any  specific  definition  for  the  value  of  a  sequence  shall  be 
briefly  designated  as  a  Z)-definition,  if  it  satisfies  the  following 
requirements: 

(i)  the  definition  gives   the  value  s  whenever  L  5„  =  5, 

n=oo 

(2)  the  definition  gives  00  whenever  Ij  5„  =  00. 

n=oo 

It  will  be  observed  that  every  definition  we  have  considered, 
either  of  summability  or  of  evaluability  (except*  Borel's  absolute 
summ-ability),  is  a  P-definition.f 

It  is  known  that  if  a  series  converges  for  every  rearrangement 
of  its  terms,  it  is  absolutely  convergent.  We  now  prove  the 
following  more  general  theorem: 

Theorem  19:  //  corresponding  to  every  arrarigement  (r)  of  the 
terms  of  a  series,  there  exists  a  D-definition  (Dr)  which  gives  the 
series  a  finite  value  Sr,  then  the  series  converges  absolutely. 

We  first  observe  that  we  may  assume  the  series  to  have  an 
infinite  number  of  terms  of  each  sign;  for  otherwise,  the  theorem 

*  Here  even  requirement  (i)  is  not  fulfilled;  see  p.  56. 

t  We  proved  the  satisfaction  of  the  first  requirement  in  all  our  cases  except 
Borel's  absolute  summability;  similar  proofs  can  be  given  for  the  second  re- 
quirement, some  of  which  are  included  in  Theorem  iia. 


74  UNIVERSITY  OF  MISSOURI   STUDIES 

is  proved,  since  the  series  cannot  in  that  case  diverge  unless  it 
diverge  to  infinity,  which  is  impossible  because  the  corresponding 
Z)-definition  would  give  oo,  thus  contradicting  the  hypothesis. 
The  series  has,  then,  an  infinite  number  of  positive  terms  («») 
and  an  infinite  number  of  negative  terms  (—  Vi).  If  each  of  the 
series 

«1   +   «2   4-   «3   +    •  •  • 
—   Vi    —    V2    —    I's   —    •  •  • 

converges,  the  sum  converges  absolutely  (for  we  could  otherwise 
find  an  arrangement  r  such  that  Dr  would  give  oo);  and  our 
theorem  is  proved.  Let  us  assume,  then,  that  one  of  the  series, 
say  the  /(-series,  is  divergent.  We  can  accordingly  choose  ki 
terms  from  the  /^-series  so  that 

A-, 

S«t    >   fl    +    I, 
i=l 

then  the  next  ki  terms  of  the  w-series  so  that 

S     Ui>  V2  +    I, 

and  so  on.     Now  consider  the  arrangement 

^Ui  —  Vi  -\-     S     W,-   —  2^2  +    •  •  • . 

The  sum  of  the  first  2«  terms  is  greater  than  n\  and  the  sum  of 
the  first  {2n  +  i)  terms  is  greater  than  w  +  a  positive  term. 
Hence  the  series  diverges  to  oo  for  this  arrangement,  and  ac- 
cordingly the  corresponding  Z)-definition  gives  it  the  value  oo, 
which  contradicts  the  hypothesis. 

A  series  may  be  defined  to  be  absolutely  convergent  in  two 
ways:  (i)  if  it  converges  when  all  its  terms  are  made  positive; 
(2)  if  it  converges  for  every  arrangement  of  its  terms.  Since 
the  concept  of  absolute  convergence  is  a  useful  one  in  the  theory 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  75 

of  convergent  series,  it  is  natural  to  ask  whether  we  can  intro- 
duce, correspondingly,  the  notion  of  absolute  evaluability  into 
the  theory  of  divergent  series.  The  two  natural  definitions  would 
be:  A  series  is  absolutely  evaluable  if  it  is  evaluable  (i)  when 
all  its  terms  are  made  positive,  (2)  for  every  rearrangement  of 
its  terms.  Consider  the  first  definition.  If  the  series  is  eval- 
uable when  all  the  terms  are  made  positive,  it  must  be  convergent; 
for  otherwise  it  would  diverge  to  00,  and  could  not  accordingly 
be  evaluable.  As  to  the  second  definition;  if  a  series  is  evaluable 
for  every  arrangement  of  its  terms,  it  is,  by  Theorem  19,  ab- 
solutely convergent.  Hence  neither  of  the  definitions  of  absolute 
evaluability  is  useful. 


§  8.   TESTS   FOR   CESARO-SUMMABILITY 

As  in  the  case  of  convergence,  it  may  happen  that  we  wish 
to  know  not  what  value  a  given  series  has,  but  whether  it  has 
any  value  at  all.  We  are  accordingly  led  to  consider  tests  for 
summability. 

We  begin  by  recalling  two  theorems  which  have  already  been 
stated : 

Theorem  :  A  necessary  condition  that  the  series  Ui  -{-  U2-{-  Uz 
+  •  •  •  he  summahle  (Cr)  is 

—  =  0.* 

n=oo  ^ 

Theorem  (3) :  A  reducible  averageable  sequence  with  a  finite 
number  of  strong  limit  points  is  Cesdro-summable  of  order  i . 

This  is  a  sufficient  condition  for  summability  (Ci).  We 
shall  now  consider  further  sufficient  conditions  for  summability 
(Ci). 

Theorem  20:  //,  in  an  alternating  series,  either  (a)  the  terms 
decrease  monotonically  in  absolute  value,  or  (b)  the  terms  increase 
monotonically  in  absolute  value,  while  the  sum  of  the  first  n  terms 
is  limited,  then  the  series  is  summahle  (Ci). 

Let  the  series  be  «i+«2+«3+*  •  •,  and  Sn  =  Ui-\-U2-\-  •  •  •+Mn- 
In  case  (a)  we  have  Sim-\  ^  •S2m+i  >  ^2;  S2m-2  ^  Sim  ^  Si.  In  case 
{h)  we  have  52m-i  ^  s^m+i  <  A ;  s^m-^  '>  S2jn>  A.  Hence  in  either 
case,  Ij  52m+i  exists  =  h;    L  52m  exists  =  1%.     By  Theorem  3, 

therefore,  the  series  is  summable  (Ci). 
As  examples,  we  may  take: 


(i)  2  -  f  +  f  -  f  + 


2 
■Sec  p.  II. 


76 


DEFINITION   OF   SUM   OF   A   DIVERGENT    SERIES  77 

(ii)        I  -h  +  &-ay  +  {iy-  •••, 

(iii)  I  —  I.I  +  I. II  —  I. Ill  +  I. nil  —  •... 

Examples  (i)  and  (ii)  illustrate  case  (a) ;  (iii)  illustrates  case  (b). 
Theorem  21:  Let 

n  _      n 

Sji    =^   ^^  Ui,       On   ^^         /  -  Si', 

00 
then  the  series  zl  ut  is  summable  (Ci)  if  either  (a)  5„<  5„+i  <  A, 

n>  N  or  (b)  Sn  >  Sn+i  >  B,n>  N. 
For 

<,     _    . ,  _  i  r  ^1  +  52  +    "  •    +  Sn-l  1  _lr  ^        , 

n\_  n  —  I  J      n 

Now  by  (a),  Sn  —  Sn-i  >  o,  and  S^  <  A.     Hence  L  5„  exists. 

Jl=00 

Similarly  for  case  (b). 

Theorem  22:  Le/  a  series zLui  satisfy  the  conditions 

(a)  the  series  is  summable  (Ci), 

(b)  \sn\  =  \'Ui  +  «2  +  •  •  •  +  Un\  <  A, 

and  let  a  set  of  positive  constants  ci  be  given  such  that  either 
(c)    Ci  >  Ci+i    or    (d)    Ci  <  Ci+i  <  A,    i  >  N;    then    the    series 
ciUi  -\-  etU2  -\-  '  •  •  is  summable  {C\). 
By  (c),  L  e„  =  ^,  and  e„  >  k. 

11=:  CO 
00 

If  ^  =  o,  2^  eiUi  is  convergent  by  a  well-known  theorem,*  and 
hence  is  summable  (Ci).  If  ^  =|=  o,  let  8n  =  Cn  —  k  >  o.  Then 
5n  ^  5„+i  >  o,  and  L  5^  =  o.     Accordingly*  the  series  2^  8iUi 

11=00  i=l 

CO 

is  convergent,  and  hence  summable  (Ci).     But  23  ^^'i  is  sum- 

t=i 
mable  (Ci)  by  (a) ;  so  that 


Sec  Goursat-Hedrick,  Mathematical  Analysis,  p.  349,  §  166. 


78  UNIVERSITY  OF  MISSOURI   STUDIES 

00  eo 

X!  (5i  +  k)lli  =  Zl  ^iUi 

is  summable  (Ci).     Similarly  for  case  (d). 
If  in  the  preceding  theorem  we  put 

00 
^Ui  =   l   —   l  +  l—  I-', 

we  obtain: 

Corollary  I :  //  the  terms  of  an  alternating  series  tnonotonically 
decrease  in  absolute  value,  the  series  is  summable  (Ci). 

This  is  Theorem  20,  case  a. 

Corollary  2 :  If  the  terms  of  an  alternating  series  remain  limited, 
and  increase  monotonically  in  absolute  value,  from  some  point  on, 
then  the  series  is  summable  (Ci). 

Since, if  \sn\  <  A,  then  |w„|  =  \sn  —  Sn-\\  ^  2A,  this  corollary 
includes  Theorem  20,  case  b,  as  a  special  case. 

Before  proceeding  to  sufficient  conditions  for  Ces^ro-sum- 
mability  of  order  higher  than  the  first,  we  shall  prove  the  follow- 
ing theorem,*  which  we  shall  soon  need. 

Theorem  23:  If  V  =  Vi  —  v^  -\-  Vz  —  v^^  -\-  •  •  •  is  an  alternating 
series  whose  terms  decrease  monotonically  in  absolute  value,  then 
the  Cauchy-product  of  V  by  the  series  I  —  i  +  i  —  i-f'-'  is 
summable  (C2). 

By  Theorem  20,  the  series  V  is  summable  (Ci);  hence  the 
product  is,  by  Theorem  (j),  surely  summable  (C3).  We  wish 
to  show  that  it  is  summable  (C2). 

{vi-  Vi-\-Vs-  v^-\-  "  •)(!  -  I  +  I  -  I  •  •  •) 

=  Vi  -  (Vi  +  ^2)  -\-  (Vi  +  V2  -{-  V3)  —   '  ■  ■ . 

The  sequence  corresponding  to  this  product  series  is: 

(a)     Vu     —  V2;    Vi  +  Vz]     —  {vi  +  v^\    {vi  +  Vz-\-  Vi);     •  •  • 

*  More  generally,  if  JJ  and  V  are  two  alternating  series  whose  terms  de- 
crease monotonically  in  absolute  value,  then  the  Cauchy-product  of  U  and  V 
is  summable  (C2).     The  proof  is  similar  to  that  given  for  Theorem  24. 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES  79 

and  the  sequence  for  Ces^ro's  first  mean  is: 

m     t'l;  -^— ; ; ;  •••. 

Let  us  write  the  odd  and  the  even  elements  of  this  sequence: 

n{Vi  -  V2)  +  (n  -  l)(V3  -  Vi)  +   •  '  •   +  {V2n-1   -  V2n) 


hn 


271 


[n{vi  —  V2)  ■\-{n  -  i)(z'3  -Vi)  +  •■• 

,  +  (Z'2n-l  -  V2n)\  +  (j'l  +  Z'3  +   •  '  '    +  ?'2n+l) 

2w  +  I 

Now  {Vi   -  V2)    +   (^^3    -  1-4)   +    •  •  •    +   (Z'2n-1    -  V2n)   +    '■  '     is     COU- 

vergent;  for  if  Sn  denotes  the  sum  of  the  first  n  terms  of  this 
series,  we  have 

Sn-i  <  5„  <  Vi,     since     y„+i  ^  z;„. 

Since  L  5„  exists, 

y      ^1    +   -^2    +    •  •  •     +   ^n 

n=oe  W 

also  exists,  i,  e., 

-P     n(Vl  -  V2)    +    («  -   OC^a   -  I'd)  +    •  •  •    +   {V2n-l   -  f2n)  X        , 

±J =     Lj2t2n 

n=oo  ''  ?j=oo 

exists.     Furthermore,  since   L  !;„  exists  (owing  to  the  relation 

11=00 

O   <  Vn+l   S   ^n), 

L  ^2n+l    =   I, 

Jl=0O 

and  hence 


Thus 


-Li  hn+l    —     ±J  hn   '  ^         I  +    J_J 


T     ^1   +  ^3  +    •  •  •    +  V2n+1   _   J 


2W  .       ^     Vi-\-V3-\ f-  V2n+l 


2n  +   I  „  =  «  W  2W  +  I  ' 

and  each  of  these  limits  exists. 


8o  UNIVERSITY   OF   MISSOURI   STUDIES 

Thus  by  Theorem  3  the  sequence  /3,  having  two  and  only  two 
limits  of  equal  weight,  is  summable  (Ci).  Hence  the  sequence 
(a)  is  summable  (C2) ;  which  we  wished  to  prove. 

If,  in  addition  to  the  hypotheses  of  the  preceding  theorem, 

Ij  i'n    =    O, 

then 

Li , =  1  =  0, 

and 

_Lj  hn+l    —   JLi  hn- 

71=00  ?i=C» 

Thus  we  have  the  theorem,  due  to  Hardy: 

Theorem  m;*  TJie  Cauchy-product  of  a  convergent  alternating 
series  whose  terms  decrease  nio7iotonically  itt  absolute  value  to  o, 
by  I  —  i-\-i  —  i+---  is  summable  (Ci). 

We  now  return  to  sufficient  conditions  for  summability. 

Theorem  24:  Let  ui  —  7/2  +  ^^3  —  «4  +  •  •  •  be  an  alternating 
series,  Ui  >  o,  and  A«it  >  o;  then  (a)  if  A^7^-  <  o,  the  series 
is  summable  (C2);  and  (Jb)  if  in  addition  J^  Aw„  =  o,  the  series  is 

n=ao 

summable  (Ci). 

Case  (a).  Consider  the  series:  Ui  —  Aui  +  A«2  —  A«3  +  •  •  •. 
Since  A«,-  >  o,  this  is  an  alternating  series,  and  since  A~Hi  <  o, 
either  A'^Ui  =  Aiii+i  —  A«,-  ^  o,  or  the  terms  decrease  mono- 
tonically.     Hence  by  Theorem  23  the  Cauchy  product 

(«i  —  A«i  +  Au2  —  Auz  +  • '  •){!  —  1  -\-  I  —  I  •  •  •) 
which  is 

=  Ui  —  (ui  +  A7/1)  +  («i  +  Aui  +  A«2)  —  •  •  • 

=   Ui   —  Uo  +  Us  —  Ui  -\-    •  •  • 

is  summable  (C2). 

Case    (b).      Here    the    series    «i  —  Aui  +  Au^  —  Aih  +  •  •  • 


*  Bromwich,  Infinite  Series,  p.  350,  ex.  9.     This  is  a  special  case  of  Theorerji 
27,  below. 

t  Atti  =  «,+i  —  m;     A"Ui  =  A(A"~'m,). 


DEFINITION   OF   SUM   OF  A   DIVERGENT   SERIES 


satisfies  the  hypothesis  of  Theorem  M,  since  the  terms  decrease 
monotonically  to  zero.     Hence  the  product  series  iii  —  7/2  +  «3 
—  «4  +  •  •  •  is  summable  (Ci). 
Thus,  for  example,  the  series 

i-(i  +  l)  +  (i+^  +  ^) 

I  -  log  2  +  log  3  -  •  •  • 

are  summable  (Ci) ;  while  the  series 

I  -  2  +  3  -  4  +  ••• 

22  +  I       3'  +  I       4'  +  I 


I  — 


+ 


+ 


234 
are  summable  (C2). 
Theorem  25:     //   in    the    series    Ui  —  «2  +  «3 

A^Mi  >  o, 

A*+i«i  <  o, 

tJien  the  series  is  summahle  (Cyfc+2);  if,  in  addition, 

Xj  A^"m„  =  o, 

71=00 

then  the  series  is  snmmable  (Ck+i)- 
Let 

I  -  I  +  I  -  •••  =  yl, 

dk  =  A^'wi  —  A*^«2  +  A'^ws  —  •  •  • , 

do  =  111  —  ih  -h  Us  —  •  ■  •• 


Then 


do  =  A{ui  —  di) 
di  =  AiAui  —  di) 


du-x  =  A{A'-hii  -dk) 
Substituting  the  value  of  di  in  the  expression  for  do, 

do  =  Aui  —  A'^iAui  —  di). 
Substituting  for  d^,  ds,  and  so  on,  in  turn, 


«i  >  o, 


82  UNIVERSITY  OF   MISSOURI   STUDIES 

do  =  Aui  -  A^Mi  +  A^Ahh  -  •  •  •  ±  A'^A'^-^ui  =f  A'dk. 

Now  dk  is  an  alternating  series  whose  terms  decrease  monoton- 
ically  in  absolute  value.  Hence  dk  is  summable  Ci,  and  A''dk  is 
summable*  (Cfc+2).  Since  do  =*=  A'^dk  consists  of  a  finite  number 
of  terms  each  of  which  is  summable  (Ct),  or  of  lower  order;  it  fol- 
lows that  do  is  summable  {Ck+2),  and  the  first  part  of  our  theorem 
is  proved. 

If  we  now  further  assume 

Ij  A^w„  =  o, 

n=oo 

it  is  seen  that  dk  is  convergent,  and  A''dk  is  summable  Ck+i. 
It  follows,  accordingly,  that  do  is  summable  Ck+i. 
*  It  can  readily  be  proved  that  A''  is  summable  (C*). 


§  9-    THEOREMS   ON   LIMITS 

The  object  of  this  section  is  to  emphasize  the  value,  from  a 
practical  point  of  view,  of  Theorem  ii,  which  we  restate  for 
the  sake  of  convenience: 

Theorem  ii:    // (i)     L  a,(w)  =  o,    for  alii, 

n—aa 

n 

(2)  Ij  2fl.(«)  =  I, 

n=ao  t=l 

(3)  either  ai{n)  >  o, 

n 

or  £lai(w)|  <  k* 

(4)  J^Sn  =  s,     or     +  oo,t 

n=oo 

then 

n 

L  ^ai{n)Si  =  s, 

or  +  00  respectively. 

We  have  pointed  out|  that  many  of  the  definitions  of  sum- 
mability  are  special  cases  of  this  theorem.  But  this  theorem 
applies  also  to  many  other  theorems  on  limits.  To  illustrate, 
we  shall  take  some  of  the  theorems  from  Bromwich's  Theory 
of  Infinite  Series.  § 

Theorem  n:    If  Bn  steadily  increases  to  00,  then 

LAn   -|-      A.n+1  An 

-W-     =      ±J     ^  _      D 

n—00  -Cn  n=«  -On+1  -Dn 

provided  that  the  second  limit  exists,  or  is  -\-  <xi. 

*  See  note  (2),  page  46. 
fSee  Theorem  11a. 
t  See  pages  43-46. 
§  Pp-  377-388. 

83 


84  UNIVERSITY   OF  MISSOURI    STUDIES 

To  apply  Theorem  ii,*  we  write: 

£>i  £>i  —  r>i-i 

ai{n)  =^;  ai{n)  = ,     i  >  i. 

Since 

n 

^ai{7i)  =  I, 
1=1 

and  since  it  follows  from  the  hypotheses  that 

Tj  fl,(w)  =  o,     and     ai{n)  >  o, 

w=oo 

we  may  apply  Theorem  ii,*  and  say:  If 

Jj  Sn  =  s     or      +00, 

w=oo 

then 

"  Ai  ^  Ai  —  Ai-\  An 

L  ^ai{7i)Si  =  ^  +  L  Zl  -^—Z — '—  =  \j-^  =  s    or    +00. 

71=00  i=l  -^1  n—<x>  i=l  -D n  n=ao  -D n 

Theorem  o:   If  the  sequences  (5„),  (/„)  converge  to  the  limits  s,  t, 
then 

LSltn  +  -^2^71-1  +    •  •  •    +  Sntl 
=   St. 

n—aa  '^ 

Here  choose  sequence 

1  /     \  tn—i+l 

Sn  =  Sn,     and     ai{n)  =  — — • 

nt 

Now 


and 


Ij  ai{n)  =  L  -  •  -  =  o 

m=ao  n=oo  n       I 


T      •^         /     X  -r      I       ^1    +   ^2    +    •  •  •    +   ^n 

Li  l^ai{n)  =  Ij =  I, 

11:=  QO     i^l  71  =  CO     I'  '^ 


«mce 

±ut„  =  t. 


*  Also  Theorem  lia. 


DEFINITION   OF   SUM   OF   A   DIVERGENT   SERIES  85 

Furthermore, 

t=i  I  n  t  n       t 

since  |  /„  |  <  k,  because 

Ij/„  =  / 

w=oo 

Hence,  applying  Theorem  ii,  we  obtain 
lu  Z^ai{n)Si=  Li  lu —  •  Si=-  Li 

SO  that 

LSltn  +  52^n-l  +    •  •  •    +  Sji 
=   S  •  t. 

We  shall  now  prove  Theorem  L,  which  we  stated  on  page  35 
without  proof. 

Theorem  l:    //  2c„  is  a  divergent  series  of  positive  terms,  then 

J     CqSq  +  CiSi  +  C2S2  +   •  •  •    +  CnSn   _    j     ^0  +  ^1  +  •^2  +   '  '  '    +  -^n 

71  =  00  ^  n=oo  n 

provided  that   the  second  limit  exists  and  either   (a)   c„  steadily 
decreases,  (b)  Cn  steadily  increases,  subject  to  the  condition 

wc„  <  (co  +  Ci  +  •  •  •  +  Cn)K, 
where  K  is  a  fixed  number. 
I  n  either  case,  we  put 

^0    +   ^1   +    •  •  •    +   .^n 

. .    a  +  i)(c»  -  g.+i)    . , 


an{n) 


^Ci 

t=0 

_  {n  +  i)Cf 

n 
2  Ci 


86  UNIVERSITY   OF   MISSOURI    STUDIES 

Since  by  hypothesis 

n 
»i  =  oo  i=0 

we  obtain 

L  a,(w)  =  o. 

n  =  oo 

Again 

T     V-        /     N         T     (C0-Ci)  +  2(Ci-C2)H \-n{Cn-l-Cr)-\-{n-\-l)Cn 

_L  Z^  «i(w)  =   Li n 

=  I. 

Furthermore,    in   case    (a),    a.(w)  >  o,    since   by    hypothesis 
Cn+\  <  Cn',  and  in  case  {h), 

n  J 

Sk»(«)l  =   li —  [(^1  -  ^o)  +  2{c<i  -  Ci)  +  •  •  •  +  «(c„  -  C„_i) 

S,""'  +  (^  +  ^)^"J 

since  by  hypothesis  c„+i  >  c„;  i.  e., 

E  kiWI  =  -^  [-  (co  +  ci  +  •  •  •  +  c„_i)  +  (2»  +  i)c„l 

t=0 


=   -  I  +  — ^i <  2  ^^ I I  <  4A. 


i=0  i=0 

Hence  in  either  case  (a)  or  (J),  we  have: 

n  J 

Li  E^iWo"!  =  L  "S: — [(^0  —  Ci)o-o  +  2(ci  —  Ci)ai  +  •  •  • 

n=M  4=0  n=oo  V~» 

^'^''  +  w(c„-i  -  C„)<r„_i  +  («  +  l)c„(r„] 

=  L  -^ [{Co  -  Ci)So  +  (Ci  -  C2)(5o  +  5i)  +  •  •  • 

n=oo  X~* 

w 

T      J     i=0     '        I  T  X      -^0  +  -^1  +    •  •  •   +  ■S^n 

=  Li        —i, \=  LiOn    =  Li 


i=0 


DEFINITION   OF   SUM  OF  A  DIVERGENT   SERIES  87 

This  theorem  is  a  special  case  of  the  following  more  general 
theorem : 

Theorem  p:    //  Xbn,  "ZCn  are  two  divergent  series  of  positive 
terms,  then 

n  n 

Lt=0  T      i=0 

n  -Li        «,  > 

l^Ci  Z^bi 

i—O  t=0 

provided  that  the  second  limit  exists  and  that  either  (a)  Cn  /bn  steadily 
decreases  or  {b)  c„/bn  steadily  increases  subject  to  the  condition 

Cn  bn 


where  K  is  fixed. 
Here  we  put 


Zl  Ci  zl  bi, 

t=0  i=0 


S  brSi 
^  n  n 


Ubi 

so  that 

bnSn   =    (^0  +  61  +    •  •  •    +  6„)o-„    —   (&0  +  &1  +    •  •  •    +  bn-l)<Jn-l, 


and  set 


an{n) 


b,  +  b,^  '•■  +bi 


Co-\-  Ci-\-  •  •  •  +  Ci  +  •  •  •  -\-  Cn 

Cnbo  -\-  bi  +    '  '  •    +  bn 


bn    Co  +  Ci  +    •  •  •    +  C„' 

In  the  first  place,  since 

n 

L  Sci  =  +  00, 

n=oo  t=0 

it  follows  that 

Ij  ai{n)  =  o. 


88  UNIVERSITY  OF  MISSOURI  STUDIES 

Also 

L  ta,{n)  =  L  -^-tir  -  r^)  (^0  +  61  +  •  •  •  +  h)  =  I. 

t=0 

Again  in  case  (a),  ai(n)  >  o;  and  in  case  (b), 
"  if 

t=o  y^       L 

+  ^  (2&0  +  2&I+  •  •  •   +2bn-i  +  b„)  1 

,       bo  +  b,+  '-'  -hb„       ^     ^^ 
=  -  I  +  2 c„  <  2K. 


Now  we  have: 


{t-ty^-'^^ii-t) 


6o5o  +  6i5i 

X  (oo+oi)     ^     ,   ^ —  + 


i=0 
+  (    7^^  ~h^)  (^O^O  +  ^l-^l'^ h^n-l5„-l) 


On 


] 


I 


—    AJ      n  ['^O-^O  +  CiSi  +    •  •  •    +  Cn-lSn-\  +  C„5„]. 

71  =  00     X"" 

i=0 

Thus  in  either  case  (o)  or  {b)  we  have  the  theorem  estabHshed, 
since 

■n 
>i  =  oo   1=0  ji=oo 

whenever  the  latter  exists. 


§  10.   CONCLUSION 

In  this  concluding  section  we  propose  to  recall  some  of  our 
main  results,  to  show  wherein  they  fall  short  of  being  complete, 
and  thus  to  formulate  the  problem  which  remains  to  be  solved. 

Our  results  of  §  3,  concerning  averageable  sequences,  are 
not  of  great  value,  since  they  involve  a  knowledge  of  the 
existence  of  certain  limit  points  before  the  question  of  the  ex- 
istence of  the  averageable  limit  could  have  any  significance. 
On  the  other  hand.  Theorem  3  is  found  useful  in  practice,  in 
showing  that  certain  classes  of  averageable  sequences  are 
summable  (Ci). 

Though  we  have  discussed  more  general  definitions,  we  shall 
confine  most  of  our  consideration  in  this  section  to  the  A- 
definition  of  evaluability. 

It  need  hardly  be  pointed  out  that  one  of  the  inadequacies 
of  the  .4 -definition  is  that  it  may  not  be  unique;  that  is,  two 
specific  sets  of  numbers  Uin  and  bin,  both  satisfying  the  condi- 
tions of  the  i4 -definition,  may  give  difTerent  values  to  the  same 
sequence.  Thus  the  sequence  Si  =  (—  1)*+^  log  i  has  two  differ- 
ent* values  for  the  two  different  definitions: 

ain=-,     bin  =-\  I  +  (-  lY+'r^.l,     i>i 
n  n\_  log  *  J 

I 
= ,     ^  =  I. 

n 

In  fact*  the  former  definition  gives  the  sequence  (5,)  the  value 
o,  while  the  latter  gives  it  the  value  i. 

Two  questions  accordingly  present  themselves.  First:  given 
two  yl -definitions,  what  is  a  sufificient  condition  that  one  defi- 

*  See  p.  38. 


90  UNIVERSITY  OF  MISSOURI   STUDIES 

nition  be  a  generalization*  of  the  other?  Secondly:  under  what 
conditions  are  the  two  definitions  equivalentf  in  scope? 

We  shall  now  consider  each  of  these  questions  in  turn.  The 
answer  to  the  first  question  will  be  made  clear  by  a  few  prop- 
ositions. 

Theorem  26:    // 

Sn    =    a:in2l  +  Q;2nS2+  •  •  •  +a;„n2„  =  6i„5i  +  &2n-^2+  •  •  •  -\-bnnSn, 

where  ain  satisfy  co7iditions  of  A-evaluability,t 

n 

2lbin=l,       OCin'>0,X        Ij  ain   =   O, 

t=l  n=oo 

and  if  Ij2„  =  s,  then  Ij5„  =  5. 

To  prove  this,  we  observe  that  by  substituting  the  expression 
for  2i  in  the  first  expression  given  for  5„,  and  equating  the  re- 
sulting coefficients  of  Si  to  the  coefficients  of  Si  in  the  second 
expression  for  5n,  we  obtain 

ainOinn  +  O-i  n-\Oin-\  n  +  C^j  n-'i.Cin-'i.  n  +    *  *  *    +  auCiin    =   bin. 

Adding  these  equations  ior  i  =  1,2,  •  •  •  w,  we  get: 

n  TO— 1 

i=j  n 

+   Oijn  X/  O,-,-   +    •  •  •    +   flln   •  flu    =    X)  bin 
i-l  i=l 

or 

ann   +   CXn-1  n   +    '  "  *    +   «;«   +    '  '  *    +   «ln    =    I- 

Thus  the  numbers  «,„  satisfy  all  the  conditions  of  Theorem  1 1 ; 
and  our  theorem  is  proved. 


*  Thus,  if  i4i  is  (G)  and  A2  is  (Ci),  then  A2  is  a  generalization  of  Ai,  if 
I  >  k;i.  e.,  if  when  Ai  gives  to  (5„)  a  sum,  then  A2  will  give  to  {Sn)  the  same 
sum. 

t  Thus  (i^r)  and  (G)  are  equivalent  in  scope;  i.  e.,  if  either  definition  applies 
to  Sn  and  gives  it  the  sum  s,  then  the  other  definition  will  also  apply  and  give 
the  sum  s. 

t  See  page  49,  including  footnote. 


DEFINITION   OF   SUM   OF  A  DIVERGENT   SERIES 


91 


Now  assuming  a„„  =1=  o,  and  considering  the  formula 

as  w  —  i  +  I   linear   equations  in  the   (n  —  i  -\-  i)  letters  q;,„, 
cti+i,  n  •••  ttnn;  the  determinant  of  the  system  of  equations  is 


ann  O 


fl^ln  '^1  n-1 

so  that 


I 


an 


—   a„„<2n-l  n-1    •  '  •   flu    =p   O, 


fltifli+1  i+1    ■  ■  ■    Ann 


dnn  0 

Qn-1   n      <2n-l   n-1 


fli+l    n       fli+l   n-1 
din  0,i  n— 1 


O  ^nn 

O  hn-\  n 


flt+1   i+l        ^i+l  n 
O'i  t+1  Oin 


D 


fltifli+l  i+1    ■  •  '   O-nn 


We  may  then  restate  the  previous  theorem  as  follows: 

Theorem  27:    //  ai„,   bin  are  numbers  satisfying  conditions 
for  A-evaluability,  and 

D 


an  4=  o,     ain  = 


an  '  '  '  ann 


>  O,*      Ij  ain  =  o; 


flwc^  •i/ 

then 

n=oo  i=l 

L    21  &in-Si  =  S. 

n=ao  1=1 

See  p.  49,  footnote. 


92  UNIVERSITY   OF  MISSOURI    STUDIES 

In  particular,  let  o,n  be  the  Cesaro  coefficients  for  (Cr), 

r{r  -\-  i)  •  •  •  (r  -{-  n  —  i  —  i) 

_    Cr+n-i-l,  n-i  (w  -i)! 

Clin 


Cr+n-1,  n-1  (r+  0(^  +  2)    "  ■   (r  -j-  U  -   l)  ' 

{n-i)\ 
so  thaton  evaluating  the  determinant  D,  we  obtain 

I   /  r(r  —  i)  \ 

ain   =   (    bin    —   rbi+l,  „  +     — bi+2,  n   •  •  ■    +  (—    lYbi+r,  n    I  , 

(la  \  1*2  / 

or,  using  the  notation 

r{r  —  i) 

{bin  —  bi+i,  n)r  =  bin   "  ^&i+l,  n  +  bi+2,  n  '  '  '    +  ("   T-Ybi+r,  n 

0!in   =  {bin  —  bi+\,  n)r=  [(^in  ""  ^t+l,  n)r-l~(^i+l.  n~0i+2,  n)r-l]. 

da  "a 

It  is  evident  that 

L  Olin    =0       if        Ij  &in    =   O; 

71=00  M  =  00 

hence  we  may  say: 

Theorem  28:  //  bin,  corresponding  to  a  definition  B  of  evaliia- 
bility,  satisfies  the  condition  {bin  —  bi+\,  „),-  >  o,*  then  if  the  sequence 
{Sn)  is  sitmmabh  {Cr),  it  is  also  evaluable  according  to  the  B-definition. 

If  we  let  bin  be  the  coefficients  for  summability  {Hr),  i.  e., 

,          ,.     ,            {i,  n)r-2   ,    {i+i,n)r-2   ,            ,   {n,n)r-2 
nbin  =  {i,  n)r-i  = : + jj— — -  +  •  •  •  + , 

where 


then 


II  I 

I       t  -j-  I  n 


{i,  w)i  -  (^  +  I,  «)i  =  -, 

(i,  w)p  -  {i  -\-  I,  7i)p  =         ■  ^- 


*  The  condition  2  \{bin  —  bi+i,  „)r|  <  K  is  sufficient. 


n(bin   —  bi+l,n)2   = 


DEFINITION    OF   SUM   OF  A  DIVERGENT    SERIES  93 

Now 

n{bin  -  bi+i,  „)i  =  [(i,  7i)r-i  -  (i  +  1,  n)r-i]  =  — — '—  , 

(i,  n)r-2  _  (t  +  I,  n)r-2  _  (i,  n)r-2  +  (i,  n)r-3 
i  i  +  I  i{i  -\-  i) 

Assume 
,  ,     _  Pi(i,  n)r-2  +  P2{i,  n)r-3+  •  •  •  +  Piji,  n)r-j-l 

n{bin  -  ^+1.  n)i-  i(^  +   I)    .  .  .    (^  +  J   _    I) 

pi  >  o. 
Then 

nibin  —  bi+i,  „)/+!  =  n  [{bin  —  bi+i,  n)j  —   (^i+1,  n  —  bi+2,  n)i] 

ipiji,   n)r-2  +  (Pl  +jP2){i,    w)r-3  +   •  •  •    +  PjjJ,    n)r-j-2 

i{i  +  i) '-•  {i-\-j) 

_  (Tlji,    n)r-2  +  (T2{i,    n)r-Z-\-    '  •  •    +  q-;+l(t,    It)  r-j-2 
i{i+l)"-{i+j) 

Oi  >  O. 

Hence  by  mathematical  induction 

,,  ,          ,    _  pi(i,  n)r-2  +  Piji,  n)r-3  +  •  •  •  +  Pi{i,  n)r-j-l 

n{bir.-bi^Un)i-  ^(^_|_i)...(i+^-_l) 

Pi  >  o,  and  accordingly  (&,„  —  Z>,+i,  „)/  >  o. 

Thus,  by  our  last  theorem,  we  may  say: 

Theorem  q:  //  the  sequence  (Sn)  is  summable  {Cr),  then  it  is 
also  summable  (Hr).* 

The  value  of  Theorem  27  is  shown  by  its  special  cases,  theorems 
28  and  Q.  We  shall  give  still  another  special  case.  Theorem  P, 
due  to  Hardy,  t 

*  This  theorem  has  been  proved  by  Ford,  Am.  Journal  of  Math.,  Vol.  32, 
1910,  and  by  Schnee,  Math.  Annalen,  Vol.  67,  1909.  The  converse  which  has 
been  first  proved  by  Knopp,  inaugural  dissertation  (Berlin,  1907),  can  also  be 
proved  by  using  Theorem  29. 

t  Quarterly  Journal,  Vol.  38,  1907,  p.  269.  Hardy  states  that  the  first 
part  of  the  theorem  had  been  given  by  Cauchy.     See  p.  87  for  another  proof. 


94  UNIVERSITY  OF  MISSOURI   STUDIES 

If  a,  >  o,    bi  >  o, 

"  " 

An   =   Jlai,       Bn   =   Y.bu        L^n   =    <»,        Ij  A  n   =    ^  ■ 

and  if  either 


i=\  n  =  a> 


or 


and  if  also 


then 


Let 


and 


bj  ^  ^i+i 
Oi  ~  ai+i 


—  <  and     -^  <  K-7- ,  K  >  o, 

Oi         a,+i  JDn  -^n 

-r      aiSi   +    •  ■  •    +   OnSn 

Li  T T ^  ^' 

n=aa      C^l   "T    *  '  '    ~r  On 
J      biSi   +    •  •  •    +   bnSn   _ 

n=«y     b,-\-  •  •  •  -]r  bn 


_    Oi^         ,        _  _^ 
Oin  A      >        Oin  -ri 


On  O 

On-1        On-l 


Ofin   '  '  '    Oi 


Oi+1        Oi+i 
Oi  Oi 


o 

bn 

o 

bn-1 

Oi+i 
Oi 

bi+i 
bi 

'Bnl'^'Ui      alxjj 


Since 

it  follows  that 

If  further 


L5„  =  oo, 

71=00 

L  ain   =  O. 


bi       bi+i 

Oi  ~  Oi+i 


DEFINITION   OF   SUM   OF   A  DIVERGENT   SERIES  95 

then  a,„  >  o.     If 

bi  ^  bi+i 

a,-  ""  fli+i' 
then 

i=i  Bn\_\a2      ax)  \az      a-i ) 

\an       an-ij  an       J 

=   ^    [-    hi    -    b.    •  '  •     -    bn-l]    +  ^   -^  {An-X   +  An) 
JJn  -D  n  an 

On   An  I         Tjr  •  On      ^    ^^      an 

=  -i+2^ <-i+  2K,     Since     ^^  <  K-  -r-  . 

Bn   an  Bn  An 

Thus  Hardy's  theorem  is  proved*  by  applying  Theorem  27.! 

Let  us  now  return  to  the  questions  of  page  89.  The  answer 
to  the  first  question  is  found  in  Theorem  27,  which  is  seen  to  give 
sufficient  conditions  that  one  of  two  definitions  of  summabiHty 
be  a  generalization  of  the  other.  Though  these  sufficient  con- 
ditions are  fairly  simple,  and  prove  useful  in  leading  to  impor- 
tant theorems,  it  would  seem  extremely  desirable  to  have  suf- 
ficient conditions  that  D  >  cj 

To  answer  the  second  question,  we  need  only  observe  that  if 
we  can  prove  by  Theorem  27  that  definition  (^)  is  a  generalization 
of  definition  {B)  and  also  that  definition  {B)  is  a  generalization 
of  definition  {A),  then  {A)  and  {B)  will  be  equivalent  in  scope. 

Now  let  {Sn)  be  summable  by  the  definition  (^4)  and  (/„)  by 
{B),  and   let   one  definition  be  a  generalization  of  the  other. 


*  The  proofs  for  this  theorem,  given  by  Hardy  (loc.  cit.)  and  by  Bromwich, 
Infinite  Series,  p.  386,  are  longer, 
t  See  p.  49,  footnote  2. 
%  See  p.  91  and  p.  49  footnote. 


96  UNIVERSITY  OF   MISSOURI    STUDIES 

Then  the  two  sequences  may  be  added  term  by  term,  and  the 
resulting  sequence  will  be  summable  by  the  more  general  of  the 
two  definitions.  For  if  A  is  taken  as  the  more  general  defini- 
tion, then  (Sn)  is  summable  by  (A)  by  hypothesis,  and  (/„),  being 
summable  by  (B),  must  also  be  summable  by  (A)  which  is  a 
generalization  of  (B).     Thus  (5„  +  /„)  is  summable  by  (A). 


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